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In several books about topology they say ''consider the plane $\Re^2$''. This bothers me, since we are considering $\Re^2$ only as a topological space. Why do we still make the geometrical analogy with the plane? Is it just for convenient reasons? If i like to make an analogy with a curved 2-d space am i wrong?

My question is since i consider $\Re^2$ to be a topological space, how should i picture it geometrically and why the picture it always as a plane? Does the topology i give in $\Re^2$ matter to how i should picture it?

Thanks

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  • $\begingroup$ Why not? If it doesn't help you to picture it that way, then picture it some other way, or not picture it at all. $\endgroup$ – Lord Shark the Unknown Oct 28 '18 at 12:14
  • $\begingroup$ but there are some restrictions i.e. its wrong to picture it as a line. $\endgroup$ – kot Oct 28 '18 at 12:36
  • $\begingroup$ @kot some properties of a topological space have a geometric interpretation like path-contentedness. The plane is rich enough to provide many important counterexamples as well. Being able to sketch subspaces or remember the spaces by the geometry can be a useful mnemonic too. $\endgroup$ – CyclotomicField Oct 28 '18 at 13:52
  • $\begingroup$ "Think of an animal!" Thirty percent of people then think of a lion. That's o.k., and the conversation can go on. "Think of a topological space!" is similar. Thinking of the set $\{1,2,3\}$ with the discrete topology in such a situation would be detrimental to the sequel of the discussion. $\endgroup$ – Christian Blatter Oct 28 '18 at 18:36
  • $\begingroup$ The plane is a useful topological space for visualizing problems, testing propositions and possible insights for proofs. $\endgroup$ – William Elliot Oct 28 '18 at 22:09
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To answer the questions in your first paragraph, we think of the plane partly for convenience reasons, but also we remember that when we talk about “topological” notions they should be independent of some of the geometric particulars.

What I mean is, while I might think of a flat plane, like a chalkboard, and you might think of a curved sheet, like a cloth or whatever, anything that I call a topological property should still be true if I were somehow able to bend or stretch my chalkboard. Maybe I can’t talk about distance, but if I draw a circle it has an inside and an outside no matter how I adjust the chalkboard (this is a surprisingly hard theorem to prove, however!)

On the other hand, you might be concerned that another, equally correct visualization of the plane exists, which you can’t tell looks like bending or stretching a sheet of paper. In this direction, we know the plane is metrizable, and has topological dimension 2, and various other facts. So I don’t think there’s any harm in thinking of it as a sheet in three-space, as long as you remember you’re allowed to play with it

Edit: Oh shoot, I meant to answer in particular your last question: No. the topology of the plane is completely separate from whatever visualization you find convenient. Which is why I would suggest, rather than emphasizing a particular viewpoint, think more about how you can change your visualization of the plane without changing the topology.

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