14  Topological Spaces

Metric Spaces ended by noticing something strange: convergence and continuity, once restated in terms of open sets, never actually mentioned the numeric distance \(d\) again. If open sets are all those ideas really need, distance itself might be more machinery than necessary. This chapter drops it.

14.1 The problem, before the definition

Take the open sets a metric produces — unions of open balls — and just keep the list, throwing away the distance function \(d\) that generated it. What’s left is a set \(X\) together with a collection of “open” subsets, and it turns out that collection alone is enough to define continuity and convergence. This is a genuine generalization, not a restatement: plenty of useful notions of “open” arise with no metric behind them at all, and it’s worth seeing one before the definition. On any set \(X\), declare every subset open — the discrete topology. Or declare only \(\varnothing\) and \(X\) itself open — the indiscrete topology. Neither needs a distance function, and the second one, in particular, has too few open sets to come from any metric: it can’t separate any two points the way a metric’s open balls always can.

14.2 The abstraction

A topological space is a set \(X\) together with a collection \(\tau\) of subsets of \(X\) (called open sets), satisfying:

1. \(\varnothing \in \tau\) and \(X \in \tau\).
2. Arbitrary unions of sets in \(\tau\) are in \(\tau\).
3. Finite intersections of sets in \(\tau\) are in \(\tau\).

14.3 Why these axioms — including the one asymmetry that looks like a typo

The first is a boundary condition: the whole space and the empty set are open by convention, so the theory doesn’t have to special-case them constantly. Closure under arbitrary unions matches the intuition that “open” means “a neighborhood of each of its points” — if \(U\) and \(V\) are both neighborhoods, so is \(U \cup V\), no matter how many sets you’re combining. The finite-vs-arbitrary asymmetry between (2) and (3) is not a typo, and it’s worth seeing why arbitrary intersections would break things. On \(\mathbb{R}\) with its usual open sets, consider the intervals \(\left( -\frac{1}{n}, \frac{1}{n} \right)\) for every positive integer \(n\) — each one is open. Their intersection over all \(n\) is \(\{0\}\), a single point — and a single point is not an open set in \(\mathbb{R}\)’s usual topology (no open ball around \(0\), however small, is contained in \(\{0\}\)). Allowing arbitrary intersections would force every single-point set to be open, collapsing the whole theory into the discrete topology and making “open” useless as a distinct notion. Restricting to finite intersections is exactly what blocks this collapse while still letting you intersect any specific, finite handful of open sets.

14.4 What follows: continuity, without a distance in sight

A function \(f: X \to Y\) between topological spaces is continuous if the preimage of every open set is open: \(f^{-1}(U) \in \tau_X\) whenever \(U \in \tau_Y\). This is not a new idea bolted onto the familiar \(\varepsilon\)–\(\delta\) definition of continuity — it’s provably the same statement, once a metric space’s open sets are the ones built from open balls. What’s new is that this version no longer mentions distance, or numbers, or subtraction at all, which is exactly why it generalizes to settings — function spaces, spaces of probability measures, spaces that show up deep in algebraic geometry — where no natural notion of numeric distance exists but a sensible notion of “open set” still does. This is the payoff Axioms and Definitions promised in the abstract: a well-chosen minimal axiom list turns out to say more, in more settings, than the richer structure it was distilled from.

14.5 What changed, one more time

What Is a Mathematical Structure? previewed this exact row: a topological space adds neighborhoods and continuity, without necessarily a distance — on top of a bare set, that’s the entire addition. Compared to Metric Spaces, a topology is a genuine generalization in the sense the map keeps returning to: every metric space is a topological space (take the open sets its balls generate), but not every topological space comes from a metric — the indiscrete topology on a set with more than one point is the standing counterexample, since it can’t separate points the way any metric space can.

14.6 What a topological space is NOT

Not automatically equipped with any of metric spaces’ extra structure — “far apart” stops having any meaning at all once distance is gone, only “contained in the same open set” remains. And, in the other direction, not every collection of subsets is a valid topology: arbitrary unions must still work and only finite intersections are guaranteed, which is easy to misapply when a specific example is chosen by hand rather than generated mechanically from a metric or an explicit closure argument.

14.7 Local versus global, made precise

This chapter is the cleanest home yet for one of the map’s five recurring threads. “Open set” is fundamentally a local notion — it talks about a neighborhood of a point. Whether a function is continuous at a point is decided entirely by what happens in arbitrarily small neighborhoods of that one point; whether it’s continuous everywhere is a global fact built by demanding the local fact at every point simultaneously. Later branches of mathematics that study global shape — how many “holes” a space has, whether it’s all one connected piece — build directly on the vocabulary this chapter just introduced, by asking questions no single open set can answer alone.

14.8 Where this leads

Structures now run from bare sets through groups, rings and fields, vector spaces, metric spaces, to topological spaces — the arc What Is a Mathematical Structure? opened. Linear Algebra is this site’s first full theory; Limits and Continuity is where the continuity definition above gets put to direct use, specialized to \(\mathbb{R}\). A later phase of this site returns to topology’s own further theory — connectedness, compactness, and the rest.