Open Sets, Closed Sets, and Topological Spaces

Topology is the most abstract chapter you will read in this book, and the most important. It strips analysis of distance, coordinates, and even continuity's $\epsilon\text{-}\delta$ definition, keeping only the bare minimum needed to talk about "nearness." That minimum turns out to be a designated collection of subsets called open sets. From this single concept, not from distance, not from coordinates, you can build continuity, convergence, compactness, and connectedness. The payoff: every theorem you prove using only open sets applies to wildly more general spaces (function spaces, manifolds, the Zariski topology of algebraic geometry, even finite spaces).

The definition

A topological space is a pair $(X,\tau)$ where $X$ is a set and $\tau$ is a collection of subsets of $X$ (called open sets) satisfying three axioms:

• (T1) $\emptyset\in\tau$ and $X\in\tau$.
• (T2) Any union of elements of $\tau$ is in $\tau$, even an infinite union.
• (T3) Any finite intersection of elements of $\tau$ is in $\tau$.

A set $C\subseteq X$ is closed when its complement $X\setminus C$ is open. Note carefully: "closed" is not the opposite of "open." A set can be open, closed, both ($\emptyset$ and $X$ are always both), or neither. Topology breaks every intuition you had about "open" and "closed" from working with intervals on the real line.

Why "finite" intersections?

The asymmetry between (T2) and (T3) is the heart of the theory. If you allowed arbitrary intersections, every singleton $\{x\}$ would be open (intersect a shrinking sequence of intervals around $x$), forcing the discrete topology on every space. The "finite" restriction is exactly what makes topology richer than that. The classical counterexample: $\bigcap_{n=1}^{\infty}(-1/n,1/n)=\{0\}$