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When trying to make sense of locales and comparing to usual topologies, I realized I have no idea how topologies relate to my everyday intuition of space. To make the question simpler, I'll restrict the question to a particular case.

Take a finite sheet of paper and deform it without cutting it. This could be represented by a map $f:[0,1]^2 \to \mathbb{R}^3$. Continuity requires that for every point and every small ball containing it in the deformed sheet has an original smaller circle inside it. This clearly is the case. But what about the converse?

It is not obvious to me at all that given a map $f:[0,1]^2 \to \mathbb{R}^3$ this condition of balls and circles will coincide with my intuitive geometric notion of deformation (even modulo stuff like self intersection, stretching, etc). This is not a precise question of course, as intuitive geometric notion of deformation is not well defined, but still I hope the question is understandable and I'd appreciate any comments on this.

This is related to How to Axiomize the Notion of "Continuous Space"? and the other questions linked there but I think there hasn't been an spelled out example so far.

J. W. Tanner
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Fernando Chu
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  • Let me offer you an example that you might like to test your intuitive notions on: define $f : [0, 1]^2 \to [0, 1]^2$ by $f(x, y) = p(x)$, where $p(x)$ is the Peano curve, a continuous surjection of $[0, 1]$ onto $[0, 1]^2$. – Rob Arthan Aug 21 '23 at 20:46
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    I don't get what you mean by "smaller circle inside it". Continuous maps can be quite counterintuitive. For example there is a map $[0,1]^2\to\mathbb{R}^3$ which has cube as its image. In other words topology allows deformation of lower dimensional objects into higher dimensional. I wouldn't say topology captures every day intuition about space, quite the opposite - it requires you to develop new intuition. It only focuses on small set of features, like general shape. – freakish Aug 21 '23 at 20:52
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    It might be helpful to consider how exactly a "cut" breaks continuity. Consider a 3-D neighborhood that contains the points "at which" the paper is torn but excludes one of the torn edges. – Ben Grossmann Aug 21 '23 at 20:53
  • @BenGrossmann That is a very good point! Thanks. Wonder if that's all the "physical" implications we can get. – Fernando Chu Aug 21 '23 at 21:08
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    Topology doesn't characterize the notion of "space" but the notion of continuity. It gives a setting where such continuity makes sense. – Jakobian Aug 21 '23 at 21:09
  • There are books about this. My intuition understands there are space and objects, and they are not the same, which may comply on notion of categories. Think on a circle as the whole available space and distances not in straight lines, but increasing as close as you come to the border. As you can see, we embedded an space within another space and set its topology to be weird-looking. regardless, it is still fullfilling within its domain. – Bruno Lobo Aug 21 '23 at 23:00
  • What is your intuition of space? In mathematics, a space is simply a set. It is additional structure that makes it a topological space or a probability space or a metric space or measure space etc. – John Douma Aug 22 '23 at 01:46

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