# how should one picture a topological space

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

• 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. – Lord Shark the Unknown Oct 28 '18 at 12:14
• but there are some restrictions i.e. its wrong to picture it as a line. – kot Oct 28 '18 at 12:36
• @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. – CyclotomicField Oct 28 '18 at 13:52
• "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. – Christian Blatter Oct 28 '18 at 18:36
• The plane is a useful topological space for visualizing problems, testing propositions and possible insights for proofs. – William Elliot Oct 28 '18 at 22:09