(How) do y’all visualize topologies?

More specifically, do you tend to have different visuals for different topological spaces all with the same underlying set?

The other day I came to the conclusion that it was weird that sets don’t automatically come endowed with some God-given topology, and this lead me to start wondering about the different topologies on the same set should look (in some vague, non-topological sense). e.g. How does R with the Euclidean topology look compared to R with the lower limit topology?

Just to clarify, I am not asking about (potential) differences in the properties of topological spaces sharing an underlying set; instead, for some fixed set X, I am curious about the fibers of the function f : {topologies on X} -> {(your) mental images}. As an example, my image of discrete R tends to be more fuzzy than Euclidean R; I imagine the normal number line, but every individual point is constantly doing a little dance perpendicular to the line in a way that suggests I could reach out and grab any single point all by itself. As another, for R with the right ray topology, I imagine the number line but “squished to the right” in the sense that if I grab it at a point and pull to the right, I’ll end up dragging the rest of the line with it (like pulling a rope) whereas if I pull to the left I can just snap a piece of it off (kind of like if you held down one side of a strand of uncooked spaghetti and then tried pulling the other side away). Currently, R with lower limit topology belongs to the same fiber as Euclidean R for me.

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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