Deleuze Guattari research

My pet peeve: 

I can't make heads or tails of Deleuze-Guattari. But I also see that they're popular and influential. So I opened a request on Manifold Markets:

A clear explanation of "A thousand plateaus" by Deleuze/Guattari with practical applied examples. | Manifold

But as yet I've not had very useful suggestions. So I asked ChatGPT / Universal Primer, and this looked more clear.

But my pet peeve stays up: Are their ideas any better than the more concrete ideas like complexity theory, and network theory? These look more useful. Aren't their ideas just cool metaphors, that sound good, but don't lead to better understanding?

Technical Breakdown:

Desiring-Machines: The most crucial aspect of their concept revolves around "desiring-machines." This notion is meant to challenge the psychoanalytic views of desire as a lack or deficiency. Instead, Deleuze and Guattari see desire as a productive force, an endless process of connecting, breaking, and reconnecting between entities. In this sense, everything can be considered a machine insofar as it is involved in these processes of production and exchange.

Assemblages: Machines are components of larger entities that Deleuze and Guattari call "assemblages" or "arrangements." These assemblages are networks or conglomerates of interconnected machines that work together to produce something, be it tangible or intangible (like social systems, ecosystems, or even ideas).

Anti-Hierarchical: Their concept also carries a fundamentally anti-hierarchical stance. Unlike traditional views that prioritize certain elements (e.g., the mind over the body, the individual over the environment), Deleuze and Guattari’s machines operate on a flat plane of consistency where everything is interconnected and of equal importance in the production of reality.

Socio-Political Critique: They use this framework to critique capitalist societies, arguing that capitalism is a highly efficient "socius" that captures desires and channels them into its own reproductive machinery, turning social and individual desires into commodities.

Analogies for Understanding:

Network of Cities: Imagine each city as a machine. Cities have inputs (resources, people, information) and outputs (products, waste, culture). They are interconnected through various means (roads, the internet, trade routes) forming a larger assemblage: the country or the global economy.

The Human Body: Consider the human body as a complex of machines. Each organ (heart, lungs, stomach) is a machine with specific functions, inputs (oxygen, food), and outputs (carbon dioxide, energy). Together, they form an assemblage that is the individual, which, in turn, interacts with other "machines" (other people, society, the environment).

Key Areas of Application:

Political Economy: Scholars in political economy have used Deleuze and Guattari's ideas to analyze the nature of capitalism and neoliberalism. They focus on concepts like deterritorialization and reterritorialization to understand global financial flows, market dynamics, and the socio-political implications of economic policies.

Organizational Theory: In the field of organizational theory, researchers apply their concepts to understand the dynamics within organizations and the broader market. The idea of assemblages is particularly useful for analyzing how organizations adapt, evolve, and interact within complex economic ecosystems.

Marketing and Consumer Research: Their concepts have also been applied to marketing and consumer research, offering a fresh perspective on consumer desires, brand communities, and the production of consumer culture. This approach views markets as assemblages of desires, goods, consumers, and cultural meanings.

Innovation and Technology Studies: Deleuze and Guattari's work on machines and assemblages has been influential in studies of technology and innovation, particularly in understanding how technological advancements and information flows reshape economic and social landscapes.

Notable Economists and Scholars:

While specific economists who directly apply Deleuze and Guattari's theories might not be widely known in mainstream economic circles, interdisciplinary scholars like Brian Massumi, Manuel DeLanda, and others have extensively engaged with their ideas in ways that intersect with economic thought. Manuel DeLanda, for example, has applied Deleuzian theory to social complexity and materialist philosophy, offering insights that are valuable to economic analysis.

Challenges and Considerations:

Interdisciplinarity: The application of Deleuze and Guattari's ideas in economics often requires an interdisciplinary approach, blending insights from philosophy, sociology, anthropology, and cultural studies.

Critique of Mainstream Economics: Utilizing their concepts in economic research often entails a critique of the reductionist, equilibrium-focused models of mainstream economics, advocating instead for a more dynamic, non-linear understanding of economic phenomena.

Analogies for Understanding:

Imagine the global economy as a vast, interwoven tapestry, where each thread represents different elements (businesses, consumers, policies, technologies). Deleuze and Guattari's concepts help us see not just the patterns on the surface but how each thread is connected, how they influence the shape and texture of the tapestry, and how changing one thread can alter the entire design.

Application in Organizational Theory:

Application: In organizational theory, Deleuze and Guattari's idea of assemblages can be applied to understand how organizations are complex systems composed of various interconnected elements (people, processes, technologies).

This application reflects their theory of assemblages as heterogeneous networks where different components interact. It emphasizes the non-linear, dynamic relationships within organizations, challenging hierarchical, top-down management structures.

Analogy: Consider an ecosystem in a forest. Just as plants, animals, insects, and the environment interact in complex, unpredictable ways, organizational elements interact within the assemblage. Changes in one part of the ecosystem can have ripple effects, similar to changes within an organization.

Application in Political Economy:

Application: Their concepts of deterritorialization and reterritorialization are used to analyze global financial flows and the impact of neoliberal policies.

These processes describe how capital, labor, and commodities move beyond traditional boundaries (deterritorialization) and then re-establish themselves within new contexts or systems (reterritorialization), reflecting Deleuze and Guattari's interest in the fluid, ever-changing nature of social and economic landscapes.

Analogy: Imagine water flowing through various landscapes, constantly moving, shaping, and being shaped by the terrain (deterritorialization). It eventually settles in new areas, forming lakes or rivers (reterritorialization), analogous to the movement and reconfiguration of economic elements in new contexts.

Application in Marketing and Consumer Research:

Application: The concept of desiring-machines is applied to understand consumer behavior, focusing on how desires are not just individual psychological phenomena but are shaped by and embedded within social and economic networks.

This reflects Deleuze and Guattari's broader theory that desires are productive forces within assemblages, challenging traditional views that see desire as lack or deficiency. It underscores the idea that consumer behavior is deeply intertwined with the social and material assemblages in which individuals are embedded.

Analogy: Think of a shopping mall as an assemblage where consumers, products, advertisements, and social interactions create a dynamic environment. Each element influences and is influenced by the others, much like desiring-machines in Deleuze and Guattari's theory, producing desires and actions in a complex web of relations.

Reflections:

By examining these applications, we see a common theme: Deleuze and Guattari's work focuses on the dynamic, interconnected nature of systems, whether they are social, economic, or psychological. Their theories challenge us to think beyond traditional categories and hierarchies, emphasizing fluidity, networks, and the production of reality through desire and interaction.

Thomas Ligotti - The strange designs of Master Rignolo 11

 In previous posts I have illustrated my favorite Thomal Ligotti story "The strange designs of Master Rignolo".  You can read, and see, the whole story here: 

Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10

But why is this story so fascinating? Let's ask Twitter and Facebook first. We'll search other sources in the next posts:

Antígrafo - @AntigrafoGris - wrote a long and cogent analysis:

TBH I never had an interpretation. I usually don't look for them, and more so in Ligotti, for whom meaninglessness is so relevant. I'm haunted by the characters' relation, conspiratorial, paranoid, despondent and opaque. It's impossible to say anything certain about them.

And even more by the painting's design. The concept of a man-made void in which one can dissolve and "inhabit", waiting, empty of thought and even sight, for the stars to blink out, is perversely alluring to me.

If you forced me to torture out some sense, I could say that R thinks that he has found a way out of existence free of any effort and suffering, but it is in itself a trap. Rignolo exists reality two times betrayed. First, it's painful and, second, it's a mockery of his design.

And the unseen forces that observe and churn beneath it all is the same universe that R tries to cheat with his landscapes. He even states that you can inhabit his paintings unobserved. A solution to the presence in the window that worries N so much, we don't know why.

The bottle at the end it's what N and G wanted before all of this started and their reward for understanding that there's no easy way out, no sense or design possible and particularly for ignoring all of it.

And maybe that's the reason we all are so conspiratorial, paranoid, despondent and opaque, because we know that it's unavoidable, but we have to choose to ignore it to avoid the worst

Serhiy Krykun - made a very good illustration - it also fits my impressions of the story:

Rignolo made me doodle it while on the verge of falling asleep. An illustration to "The Strange Design of Master Rignolo" by Thomas Ligotti.

Nicole Cushing - @NicoleCushing:

"...countless stars stared down like the dead eyes of sculptured faces."

"The Strange Design of Master Rignolo" by Thomas Ligotti in NOCTUARY, magnificently dark.

And then I discovered that I myself had already made some illustrations for the story, based on real photographs of Rotterdam:

It was well into evening and for some time Nolon had been been seated at a small table in a kind of park. This was a long, thin stretch of land - vaguely triangular in shape, like a piece of broken glass - bordered by three streets of varying breadth, varying evenness of surface, and of varying stages of disintegration as each thoroughfare succumbed in its own way and in its own time to the subtle but continuous movements of the slumbering earth below. ... There were other tables here and there, all of them unoccupied, but most of the park was unused ground covered with a plush, fuzzy kind of turf.

" ... I was out in that field, the one behind those empty buildings at the edge of town where everything just slides away and goes off in all directions. And there's a marsh by there, makes the ground a little, I don't know, stringy or something. No trees, though, only a lot of wild grass, reeds, you know where I mean?"

“The face,” he said, leaning back in his chair. “It was right there, about the size of, I don’t know, a window or a picture hanging on a wall, except that it was in the ground and it was a big oval, not rectangular in any way. Just as if someone had partly buried a giant, or better yet, a giant’s mask."

Ian Davey - @unapersson:

Two men visit a landscape artist whose work has an otherworldly life of its own, and they help him reach its source.

Cyclonopedia by Reza Negarestani

 

I had a chat about Cyclonopedia with ChatGPT. We agree on a lot of things. It's a difficult but fascinating book:

The book is an ambitious attempt to use philosophy and literature to challenge conventional wisdom about the role of oil in contemporary society and to offer a new way of understanding the relationship between politics, economy, and the environment.

To fully understand the philosophical ideas presented in "Cyclonopedia," some background in continental philosophy, particularly the works of Gilles Deleuze and Félix Guattari, may be helpful. However, one can still engage with the book's ideas without a deep knowledge of philosophy, as long as they are willing to approach the text with an open mind and a willingness to grapple with complex ideas.

There are at least three layers of narrative in "Cyclonopedia." The first layer involves the narrator's search for the writings of the archaeologist, which serves as the overarching framework for the book. The second layer is the narrative of the archaeologist himself and his research on the connections between oil, culture, and history. And the third layer is the presentation of the archaeologist's ideas, which are interspersed throughout the book in the form of theoretical reflections, philosophical speculations, and allegorical fictions.

In addition to these three layers, there are also other narrative threads that run throughout the book, including the themes of horror and science fiction, the use of metaphor and allegory, and the exploration of Middle Eastern politics and culture. These threads are intertwined with the main layers of the narrative, adding depth and complexity to the book's overall structure.

The book presents a vision of the Middle East as a place where the lines between the past and the present, the natural and the supernatural, and the human and the non-human are blurred. The author argues that this "haunting" is not simply a metaphor, but is a real and tangible phenomenon that has shaped the region's history and politics.

From a strange planet - 55

 I play around with AI:

The view from the traffic webcam at E39 Vaulekrossen in Norway shows a busy stretch of highway with cars and trucks moving in both directions. The road is surrounded by lush green trees and hills in the distance. There are also several buildings and structures visible along the roadside, including a gas station and a few houses. The sky is clear and blue, with a few scattered clouds. Overall, the scene is peaceful and picturesque, with a steady flow of traffic moving along the highway.

I will describe a scene: A road, behind that a small lake, behind that a hill. I see a small house near the lake and a huge farmhouse higher up the hill. The hill is grassy and stony. It is winter and the trees along the road are bare. The sun is shining and the sky is blue. Please guess what story could be placed here.

Based on the description of the scene, it seems like a peaceful and serene setting. The presence of a small house near the lake and a large farmhouse higher up on the hill suggests that there may be people living in the area. The fact that it is winter and the trees are bare could indicate that the story takes place during the colder months of the year. The blue sky and shining sun also add to the sense of calmness and beauty in the environment.

One potential story that could be placed in this scene is a tale of a young family who has recently moved to the small house near the lake. The father works on the huge farmhouse higher up the hill, while the mother stays at home with their young children. As the winter progresses, the family faces challenges and struggles, but ultimately they come together to overcome their obstacles and find happiness in their new home.

From a strange planet - 48

We still follow the E39. That's one way to not get lost in the myriad landscapes that the cameras reveal to us. We will see how far we get. There are many sights to discover. The most spectacular is the brooding house. But we also have silent lakes, dark woods and the ball in the hills.

The quotes are from this book:

Lacan at the Scene (mit.edu)

Lacan at the Scene by Henry Bond | Goodreads

E39 Vaulekrossen 1 - Streetview

The staring house. The gaze of the Other

And this detail of a darkened window, or a house that looks on, has been elaborated in turn by a sequence of  influential philosophers including Sartre, Merleau- Ponty, Lacan, and Žižek, each of whom has invoked it in relation to the notion of the Other’s gaze - for Žižek it is Norman Bates’s home in Hitchcock’s Psycho, while for Sartre it is an - inexplicably poignant - white farmhouse at night during war- time.

In his chief commentary on the theme of the Other ... Sartre offers several examples of objects that can come to embody the Other’s look. He notes how the Other’s gaze will be given just as well on occasion when there is a rustling of branches, or the sound of a footstep followed by silence, or the slight opening of a shutter, or a light movement of a curtain. ... the actual eye of the watcher, hidden behind the curtain, behind a window in the farmhouse. [Source: Lacan at the Scene by Henry Bond]

Lacan comments: “all that is necessary is for something to signify to me that there may be others there ... [A] window if it gets a bit 

dark, and if I have reasons for thinking that there is someone behind it, is straightaway a gaze. From the moment this gaze exists, I am already something other, in that I feel myself becoming an object for the gaze of others.

Slavoj Žižek has also isolated the motif of a house that looks on, asking: Is this notion of the gaze not perfectly rendered by the exemplary Hitchcockian scene in which the subject is approaching some uncanny threatening object, usually a house? There we encounter the antimony between the eye and the gaze at its purest: the subject’s eye sees the house, but the house - the object - seems somehow to return the gaze ... this gaze is effectively missing, its status is purely fantasmatic.

This image of “house” also has the quality of a Jungian archetype - a communal (society-wide) unconscious foreknowledge. For example, when young children draw, they often begin with a depiction of a face or a house - in front elevation—and these two images seem to coincide, with the mouth and eyes in the same formation as the windows and door.

E39 Søyland - Streetview

No that's not a side road into the hills. It just looks like that. 

There are just flat fields on the right.

And that's like me, trying to understand the surroundings.

Looking for mysterious surroundings, 

like this mysterious lake, just around the corner.

Rv. 444 Torger Carlsensgate 1 - Streetview

At first this landscape does not look to promising.

But there are beautiful wide landscapes with many scales of depth.

And dark woord at the end of the road.

Fv. 510 Rege - Much more details in an earlier post

There is a sphere in the hills. And a church.

You can see them in detail by following the link above.

The hills at sunset.

The hills at midday.

From a strange planet - 32

Church webcams at night. Wonderful and awesome!

"One moment of worship is worth more and spiritually more fruitful than the most busy work." - Pope John Paul II

During the winter holidays of 2020 I discovered the Irish church webcams. It was late at night after a long evening of browsing. I was immediately captivated by the spiritual atmosphere of these silent and calming pictures. Dark spaces lighted only by the altar lamp. The visible presence of Christ. Here I invite you to visit these places, especially at night.

 St. Patricks Church - Church Media live streaming

In Catholic churches, the lamp of God is an oil lamp near the tabernacle that stays on as long as the Holy Sacrament is present in the tabernacle. The lamp of God bears witness to Christ's True Presence. Most God lamps have a red glass, to distinguish the lamp from other votive lights in the church.

St. Columcille’s Pastoral Area Oratory

The beauty of Eucharistic Worship is the atmosphere of a secret that is approached not with reason, but only with the heart, which is addressed by symbolic language. The atmosphere around eucharistic worship is one of benevolent silence and warm intense security. We are invited to bask in the sun of God's love. Remember the word of Jesus: "the Father in heaven who raises his sun over the wicked and the good, and makes it rain over the righteous and the unjust" (Mt 5).

St. Mary’s Church

From a strange planet - 20

A love poem for E8 Grønbakken.

There is a glow.

The glow reminds me of Thomas Ligotti.

Everything reminds me of Thomas Ligotti.

The glow is outside and not inside.

Is that a problem?

In the evening the glow grows.

It gets stronger.

It gets bigger.

It occupies the area.

It erases and blinds.

It makes white. It makes snow.

There is a curve in the road.

At night other lights appear.

They have sparks.

They speak with the glow.

There are poles.

Two poles, one pole.

They stand in the middle.

They do nothing.

There is snow.

Hey snow, why do you cover everything?

Why are your mounds so round? So light?

Why is the shed so straight? So dark?

There is a curve in the road.

Hey curve, why do you change direction?

Why is your arc so round?

Why is the shed so straight?

There is snow.

There is a mountain in the distance.

Why is it far?

Why is it there and not here?

It is far because it looks like the surface of Mars.

A Martian mountain.

The snow melts.

Goodbye snow, goodbye winter.

I hate to see you go.

Before there was light.

Now the road turns dark.

Now the curve turns dark.

The shed likes that.

There is a one big shed.

And there are two small sheds.

And a house in the background.

They talk.

And listen.

And wait.

For winter.

For snow.

There is dust.

There is sun.

The dust is cold. The sun is cold.

It's Martian dust.

The background is green.

There is life on Mars.

There are potholes.

They turn to puddles when it rains.

When there is no rain, there are no puddles.

But the potholes are still there.

Now invisible. But present.

And waiting.

Waiting for rain.

Waiting for snow.

On Mars there is enough time to wait.

Topology for artists - workshop sketch 11

Topology for artists, walkers and other non-mathematicians

One of my little dreams is to make and give a workshop about topology for artists, walkers and outher outsiders. It might be called: The topological toolkit. What you didn't know about spaces. Or maybe: Topology as philosophy of space. These are preliminary notes: YouTube topology, Styles of topology, Spaces and maps, Translation, Words in linguistic space, Workshop contents sketch, Workshop content sketch 2. I try to post small updates often, to keep myself working on this.

---

I found this original book in the Rotterdam library. It's called 'Topologie door zien' that translates into 'Topology through seeing' or 'Intuitive understanding of topology'. It is a hand illustrated book. The author took drawing lessons to make the pictures. It's an excellent illustration of the weirdness of topological objects. Many of the illustrations are eerie. I don't think it was done on purpose, but the illustrations are just as uncanny as the Codex Seraphinianus. If I make a workshop, I would like to make that the central point.

The Antoine wild sphere - mapping a sphere to the Cantor dis-continuum. This has something to do with the Antoine necklace, another wild object.

Continuity - as an introduction to homeomorphism.

The Alexander wild sphere - as an example of a pathological object.

A proof of the hairy ball theorem.

The Warsaw circle and the Knaster bucket handle.

The Sierpinsky carpet, containing both the Knaster bucket handle and the Warsaw circle.

Compactness - contrasted with closed-ness.

Also found this:

https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-antoines-necklace/ - antoine necklace is a cantor set
http://www.carliner-remes.com/jacob/math/project/ - geometric toplogy
https://people.math.osu.edu/fiedorowicz.1/math655/ - geometric topology
https://jeremykun.com/2012/08/26/metric-spaces-a-primer/ - metric spaces
https://www.quora.com/How-can-I-show-that-the-taxicab-metric-space-and-the-standard-Euclidean-2-space-induce-the-same-topological-space - metric space and toplogy

Topology for artists - workshop sketch 7

Topology for artists, walkers and other non-mathematicians
One of my little dreams is to make and give a workshop about topology for artists, walkers and outher outsiders. It might be called: The topological toolkit. What you didn't know about spaces. Or maybe: Topology as philosophy of space. These are preliminary notes: YouTube topology, Styles of topology, Spaces and maps, Translation, Words in linguistic space, Workshop contents sketch. I try to post small updates often, to keep myself working on this.
I browsed XKCD for "topology" I found a few discussions about organizing a workshop for beginners. See the list of sources at the end.

---

Encouragement for making a workshop
I feel encouraged by this statement. And I still don't know if this has anything to do with the author of the XKCD comic (I guess not, but I'm not sure):

Topology is fun to talk about, since people who don't actually study it have never heard any of this.

I'd like to take this opportunity to encourage any and all grad students to organize their own student seminars. The best way to really learn a topic is to give a talk about it, to force yourself to be able to explain it in understandable terms. Also, the best way to become comfortable with speaking is to give lots of talks. There are many things I'd like to improve about my speaking style, but I think it's safe to say that anything good about it is a result of the many talks that I gave in this seminar over the course of my graduate career.

Some items will be difficult to translate into "normal" (non-mathematical) concepts. I indicate this difficulty with 1 to 3 stars.

Start from our normal space *
Skimp the technical stuff for conceptual understanding. You can't teach topology in an hour, but you can get the students interested in it in that time frame.

Start with a concrete example. We don't need to talk about the general case of a topological space, and it would be pointless to try to do so in an hour-long presentation. One- and two-dimensional space is something everyone is familiar with, so take advantage of that. The entire concept of open and closed sets comes directly from observations in the metrics on R and R^2.

Experiment with spaces *
The intro I like to talk about is gluing together edges of a square in order to make a sphere (all glued into a single point), a torus (opposite sides glued), a Klein bottle (one pair of opposite sides glued with a flip), and real projective space (both opposite sides glued with flips).
What do a sphere, a torus, and a Klein bottle all have in common (but a plane doesn't)?

Open and closed sets *
I think it might be good to show using simple, concrete examples. First, explain what open and closed sets are, in nontechnical terms, and give examples:

• An open set is a set where every point can be wiggled a little bit and still be inside the set
• A closed set is a set whose complement is open
• Examples of open sets in R include (0, 1), (-pi, pi), and (0, infinity)
• Examples of closed sets in R include [0, 1], [0, 42], [-10, 10]
• Open and closed are NOT opposites. The interval (0, 1] is an example of a set which is NEITHER open NOR closed. The empty set and the set R itself are BOTH open AND closed.

Structure of space - open sets as building blocks **
A topology is a set along with a collection of open subsets of that set. A topology completely specifies the topological properties of its underlying space. The underlying space is just a convenient metaphor; it is not actually necessary.

The topology doesn't just 'determine' the open sets; it 'is' the open sets. A topology on a set is just a collection of subsets that is closed under arbitrary unions and finite intersections, and which contains the empty set and the whole set. It has no more structure than this. Every topological concept must be definable *entirely* in terms of basic set/logical operations and whether or not certain sets are members of the topology, or it can't be purely topological.

A topology on a set is just a sort of rule telling us which points are 'close together'. It does this by specifying something called a closure operation: the closure of a set A is the set of all the points 'close' to A. Since the idea of topology is to study this idea in general, though, we'll let any operation count as a closure operation, and just redefine 'close' to match, as long as it satisfies a few basic rules: (list the Kuratowski closure axioms).

The hard thing to get used to about topological spaces is just how little structure is imposed by the axioms. For instance, the topology arising from an arbitrary metric space is already much more special and well-behaved than an arbitrary topology.

Neighborhoods *
Go on to explain these concepts generalize into R^2 naturally.
A circle including all points inside and including the boundary is an example of a closed set. A circle including all its interior points, but without the boundary is an example of an open set.
An open set containing a point p is called an (open) neighborhood of p.

Metric spaces and distances ***
If you're talking about distances, you're in a metric space. In what's called a metric space, for example, an open set is one where every point inside has a small region around it that's still inside the set.

Closed versus bounded *
Closed and bounded are unrelated, in the sense that we can find examples of subsets of topological spaces for every combination of {closed, not closed} and {bounded, not bounded}. In fact, we can find subsets of all four types just within the real line:

• [0,1] as a subset of the real line is closed and bounded, and its complement (-infinity,0) U (1,infinity) is neither closed nor bounded.
• (0,1) as a subset of the real line is bounded but not closed, and its complement (-infinity,0] U [1,infinity) is closed but not bounded.

Mappings and functions ***
Then, it is nice to produce a useful result with the definitions. The topological definition of a function is a really good example.
A function f maps points from R^2 to R^2. We can also think of f mapping subsets of R^2 to subsets of R^2 in a natural way: If X is a set, then f(X) is the set {f(x) for x in X}.
Then, finally, a continuous function from R^2 to R^2 is one where the following is possible. Given any open set in R^2 called Y, we can find an open set in R^2 called X such that f(X) is a subset of Y.
The reason for using R^2 is it makes for a nicer picture. Wikipedia has a nice picture of what I mean (though, the names of the variables are different): en.wikipedia.org/wiki/Continuous_function

Pathologies ***
Finally, some pathology: you could display the multitude of limits of a sequence in a non-T2 space etc. If you really want to, you can talk intuitively about the long line (although I doubt there's anything intuitive about it ).

If your audience is at a high enough level, it might be fun to show some "strange" examples. Define a homeomorphism and talk about why we require a continuous inverse to a continuous mapping (ask them for an example; if they've only ever seen real functions of a real variable, they'll have to think about it), show them some homeomorphic spaces (specifically do the stereographic projection of a sphere onto a plane), maybe show some manifolds (they look nice; it might be worth it to mention fundamental polygons here, at least for the torus and the Möbius strip, perhaps the projective plane).

Further study ***
https://en.wikipedia.org/wiki/Space_(mathematics)

Sources
https://etale.site/xkcd/
http://forums.xkcd.com/viewtopic.php?t=29671
http://forums.xkcd.com/viewtopic.php?t=37039
http://fora.xkcd.com/viewtopic.php?t=66742
http://forums.xkcd.com/viewtopic.php?t=111798

Topology for artists - workshop sketch 5

Topology for artists, walkers and other non-mathematicians
One of my little dreams is to make and give a workshop about topology for artists, walkers and outher outsiders. It might be called: The topological toolkit. What you didn't know about spaces. Or maybe: Topology as philosophy of space. These are preliminary notes: YouTube topology, Styles of topology, Spaces and maps, Translation, Words in linguistic space. I try to post small updates often, to keep myself working on this.

I read a lot of discussions on topology. Below is a summary of what I learned, structured as a workshop outline. See the list of sources at the end.

---

The nature of space

Topology studies the nature of the space itself. Does it have holes? Is it close together (compact) or spread out? This allows higher math to define calculus over strange spaces.

For workshop:

• Think about the difference between the sphere and the torus.
• Discuss the possibility of living in finite but unbounded space (asteroids game). 

(Point set) Topology as basic theory

It's like calculus, linear algebra, set theory or most other fields of mathematics that are taught in college or the first year of graduate school. You have to know them because you'll be using them in whatever you will be doing as a mathematician. You probably won't be doing research within one of these fields, but they're all used in current research.

I’m not aware of any applications of pure point set topology. Usually, applications involve more complicated topological structures, like manifolds or knots. However, if you want to really understand these more complicated structures, you need to understand the more basic theory, particularly because point set topology shows you just how unintuitive general topology is. In the general setting, there are all sorts of weird counterexamples that don’t feel like they should exist, but you need to be aware of these if you want to have the right intuition of how much information you need to be able to solve various problems. 

For workshop:

• Discuss the extremely basic concepts introduced in topology, that demonstrate that "things could have been totally different and very weird" (like the multiverse).
• Try to show some weird spaces, very different from ours (maybe using separation properties).

Structures on a space

Point set topology is the basic study of topological spaces. A topological space is sort of the minimal amount of structure that you need in order to be able to talk about convergence, connectedness, and continuous functions.

Most modern mathematics concerns "sets with structure" of various sorts. The problem is that the structure of "topological space" by itself isn't really strong enough to produce much interesting mathematics--topological spaces by themselves are too wild to make many interesting statements that hold for all topological spaces. Point-set topology studies potentially pathological topological spaces from an essentially set-theoretic point of view.

For workshop:

• Think about the building blocks of our daily space.
• Discuss bases.
• Discuss the standard (open balls) and the discrete (points) topology on a space.
• Discuss how different topologies lead to different and weird properties of a space (Sorgenfrey line is much too complicated for workshop).
• Try to present a few "wild" spaces (if possible).

Geometry without a metric

Topology seems to have arisen to answer the question "What is geometry without a metric, without a way of measuring distance?" It's a study of geometry where there isn't a metric -- a way to measure the distance between two points. Several lines of thought from different areas converged into questions about the nature of the space itself.

For workshop:

• Think about the different practical metrics in our daily (Euclidean) space.
• Discuss measuring distance in kilometres versus measuring distance in time or in difficulty.
• Discuss the difference between car, bicycle or walk distance.
• Discuss manhattan-block distance (pixels).
• Discus distance through and on a sphere.
• Discuss the space of music or literary styles.

Closeness, nearness

In the broadest sense, topology formalizes the notions of nearness and continuity. Reality appears to behave in a mostly continuous manner, at least at the scale on which humans and human intuition evolved. We tend to think of things in a continuous manner, we tend to interact with reality in a continuous manner, and now we have topology to formalize all of these continuous intuitions and interactions.

It is like geometry in that studies space, but it does not worry about the nuisances of angles and distances (even if metric spaces are touched on, which do use the latter), instead simply speaking of “closeness” (adjacency).

For workshop:

• Think about continuity and discontinuity in our daily surroundings.
• Discuss the absence of infinities, and infinite gradients, in daily experience.

Open and closed

Topology (point-set or general topology, that is) fundamentally captures the idea of continuity in a general setting, via open sets (or closed sets, if you like) which remain open after pullback.

For workshop:

• Think about open and closed sets.
• Discuss the openness or closedness of concepts like "marxist" or "city centre".

Homeomorphism

Topology is also cool in its own right. It is fascinating that we can rigorously consider objects continuously deformable into each other to be equivalent (c.f., homeomorphism & homotopy equivalence) and get something out of that. I am sure you have heard of the donut and the coffee cup, which gets the vague idea across.

For workshop:

• Think about mappings between different domains.
• Think about difference between map and reality. 

Countable and uncountable

For workshop:

• Countable and uncountable sets (maybe mention Cantor and infinities).
• You can cover an uncountable space with a countable set of open sets (I think).
• How to make this practical? I find Q versus R fascinating, but is it explainable to outsiders?

(Point set) Topology as a mature subject

Point-set topology is a “dead” field the same reason classical real variables or elliptic integrals are: the subject has been studied extensively, the hidden nuggets have been extracted, and further study of it is largely considered uninteresting. Everything in it has already been studied and proven.

Point set topology is basically a dead field, and is really not a popular research topic. For most interesting problems that use topology, it is necessary to assume spaces are nice enough that stronger techniques--such as those from algebraic topology, differential topology, or functional analysis--become available. Certainly point set topology forms a foundation for a lot of really interesting mathematics, but it isn't really studied much for its own sake anymore.

For workshop:

• Since 1900 mathematicians have thought about the structures and properties of space. This was done to produce a solid theoretical underpinning of "normal" mathematics.
• Topology is more philosophical than practical. It presents us with puzzles and makes us think.

For further study

You must be comfortable with concepts like compactness, connectedness and homeomorphisms. You also need partitions of unity, subordinate to an open cover (very important!) and for this you have to be familiar with (something like) the smooth Urysohn lemma. Finally, some understanding of quotient topology will be a plus, since it gives many important examples (Klein bottle, Projective space).

For workshop:

• Find the connection between point set topology and the weird structures of the klein bottle or the projective plane. I haven't found that yet.

Sources

Importance of topology:

• https://www.quora.com/What-is-the-philosophical-meaning-of-point-set-topology
• https://www.quora.com/How-was-the-field-of-topology-developed
• https://www.quora.com/How-important-is-point-set-topology-for-a-mathematician
• https://www.quora.com/What-would-be-a-laymans-definition-for-point-set-topology
• https://www.quora.com/How-much-point-set-topology-do-I-need-to-study-manifolds
• https://www.quora.com/What-is-a-point-set
• https://www.quora.com/Why-is-topology-important
• https://www.quora.com/How-useful-would-an-independent-course-in-point-set-topology-be-for-studying-more-rigorous-mathematical-analysis

Non importance of topology:

• https://www.quora.com/Why-is-point-set-topology-a-dead-field
• https://math.stackexchange.com/questions/2111408/is-point-set-topology-a-dead-field
• https://www.quora.com/Do-people-still-do-research-in-Point-set-Topology

For later research:

• https://www.quora.com/What-are-the-greatest-lols-in-topology
• https://www.quora.com/How-is-topology-used-in-theoretical-computer-science
• https://cstheory.stackexchange.com/questions/2898/applications-of-topology-to-computer-science
• https://staff.fnwi.uva.nl/j.vanmill/papers/papers1990/opit.pdf
• http://pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf
• http://www2.math.ou.edu/~amiller/4853/TopologyNotes.pdf
• https://mathstrek.blog/2013/01/18/topology-basic-definitions/