What is it, intuitively, that makes a structure "topological"?

Question

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?

Answer 1

• A topology tells you which things are in the vicinity of one another.
• A topology captures the idea of proximity in a qualitative way.
• A topology tells you which things are in the vicinity of one another (characterized at points being in an open set are in the vicinity of one another.)

• From the idea of proximity, you can construct definitions of other concepts such as:

  • Continuity, a property of functions such that moving to a nearby point in the domain will always cause you to move to a nearby point in the codomain.
  • Connectedness, a property of spaces which means that any way you divide your space into neighborhoods will have some neighborhoods overlap.
  • Compactness, a property related to being "finite in extent": any way you divide your space into neighborhoods, you can find a finite number of neighborhoods containing all of the points in the space.
  • Cutting and gluing, which are functions that change the topological structure of a space. Cutting separates points that were previously nearby, and gluing creates nearness relations that previously didn't exist.
  • Convergence and limits, a property of sequences converging to a point such that everywhere in the vicinity of the point, you can find the tail of the sequence.

  And so on. These are topological definitions because they are completely defined in terms of proximity (open sets).

• As for the axioms of topology:

  Imagine having rulers of various lengths, and the ability to test whether an object is approximately the same length as the ruler. With this ability, you can also devise compound tests for whether an object matches any one of several rulers, or whether it matches all of those rulers simultaneously. If the set of rulers is infinite, you can still easily prove that you match at least one of the rulers—by exhibiting one of the matching rulers. On the other hand, if the set of rulers is infinite, you may have to produce an infinitely long proof if you want to show that it matches all of them.

  These approximately-matching rulers represent the topologist's idea of open sets. Accordingly, this story motivates the definition of topologies as closed under arbitrary unions (testing whether any ruler approximately matches) and only finite intersections (testing whether simultaneously all rulers approximately match.)

• By formally capturing ideas about proximity, topologies enable us to reason about important related ideas involving shapes, holes, connectedness, smoothness, boundedness, and so on. The abstract definition of topologies means that we can reason about nearness even when the objects we're talking about are extremely nonphysical, such as sentences and structures. We can export our intuitive understanding of topology to aid in understanding highly abstract structures.