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- Intuition behind topological spaces 3 answers
With most concepts in mathematics I've learned about, I have an intuitive feeling of what their meaning is, and why we have them.
Topology is not one of them.
I understand that the concept of a topology emerged from the study of continuous functions. Essentially as far as I understand it, the starting point historically was the epsilon delta definition of continuity on real functions, and the topology axioms are "what remained after stripping away everything not required for the definition of continuity".
So far so good. But then it turns out that various mathematical structures have topologies defined on them, while these have (seemingly to me at least), absolutely nothing to do with continuity, such as topologies defined on sets of first-order sentences, or on "sets of mathematical structures" (in mathematical logic), or on any kind of other set which as far as I see has no relation to $\mathbb R$.
Of course what these all have in common, is the topology axioms. But while I know what the axioms are and can apply them, I have no intuitive understanding of what it is that all these different structures have in common.
What, intuitively, does it mean for a structure to be "topological"? I intuitively know what the set of vector spaces have in common, or the set of measure spaces. What , intuitively, does the set of topological spaces have in common?