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  • $\begingroup$ John Stillwell suggested in the preface to his wonderful-if highly unorthodox-textbook on the subject that to render topology more concrete, that continuity should be removed entirely as it's foundation and it should be expressed entirely in terms of finite elements like simplexes.I think that would really work only in low dimensional spaces that are locally Euclidean. My point is that alternatives to open and closed sets (and the more common "weaker" variant nieborhoods) have been proposed in the past. $\endgroup$ Commented Mar 24, 2010 at 17:34
  • $\begingroup$ > Then the rest of the definitions are just like metric space definitions without the need to reformulate those definitions in terms of open sets. … [A] function $f : X \to Y$ is continuous at $x$ if and only if the following condition holds: for every b.o.n. $M$ of $f(x)$ there exists a b.o.n. $N$ of $x$ such that $f(N) \subset M$. I think that I'm missing something very important here. This seems like you have re-formulated the metric-space definitions, just saying ‘b.o.n.’ everywhere the usual definition would say ‘open containing’. What is the simplification? $\endgroup$ Commented Mar 24, 2010 at 21:38
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    $\begingroup$ Here's an example. Suppose I define the product topology on X={0,1}^N. I won't tell you what an open set is. Instead I'll say that the b.o.n. B_n(x) is the set of all sequences that agree with x up to n. Then I can define the continuity at x of a map from X to X without ever having to say what an open set is. So basic open neighourhoods are playing a role similar to balls of radius epsilon in metric-space theory. To relate this definition to metric spaces I don't have to prove that a map between metric spaces is continuous if and only if the inverse image of an open set is open. $\endgroup$ Commented Mar 26, 2010 at 9:42