Topological Spaces & Open Sets

The three axioms a collection of open sets must satisfy.

The idea

A topological space is a set carrying just enough structure to support the notions analysis is built on — limits, convergence, continuity — without any notion of distance. The structure is a choice of which subsets to call open.

The definition abstracts the open sets of the real line. There, a set is open when around each of its points it contains a whole interval. These sets have three properties, and the basic theorems use nothing else: the empty set and the whole line are open; any union of open sets is open; the intersection of finitely many open sets is open. The finiteness matters: each interval $(-\tfrac{1}{n}, \tfrac{1}{n})$ is open, yet their common intersection is the single point $\{0\}$, which is not. The figure shows the first three of these intervals closing onto $0$.

So keep the three properties and discard the distance. A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ satisfying exactly those three rules. Its members are the open sets of the space, and the pair $(X, \mathcal{T})$ is a topological space. Openness is no longer a property a set has on its own: a set is open exactly when it is listed in $\mathcal{T}$, and choosing that list is what it means to give $X$ a topology.

Ways to work on it

• Walkthrough. The topology axioms, tested on small collections.
• Practice. Decide whether a candidate collection is a topology.
• Hardest. Repair a collection and name the failing axiom.

Not sure where to start? Take the ten-question placement test.