Topological Space

Let $X$ be a set.

Definition. A topology on $X$ is a collection $\tau \subseteq \mathcal{P}(X)$ satisfying:

1. $\emptyset \in \tau$ and $X \in \tau$.
2. If $\{U_i\}_{i \in I} \subseteq \tau$, then $\bigcup_{i \in I} U_i \in \tau$ (closure under arbitrary unions).
3. If $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$ (closure under finite intersections).

The pair $(X, \tau)$ is a topological space. Members of $\tau$ are called open sets.

Proposition. On any set $X$, there is a coarsest topology $\tau = \{\emptyset, X\}$ (the indiscrete topology) and a finest topology $\tau = \mathcal{P}(X)$ (the discrete topology). The collection of all topologies on $X$, ordered by inclusion, forms a complete lattice.

Proof sketch. An arbitrary intersection of topologies on $X$ is a topology on $X$ (each axiom is preserved by intersection). The lattice join of a family of topologies is the coarsest topology containing all of them, which exists because the intersection of all topologies containing the family is itself a topology. $\square$

Proposition. The open sets $\tau$ of $(X, \tau)$, ordered by inclusion, form a frame: a complete lattice in which finite meets (intersections) distribute over arbitrary joins (unions). Equivalently, $\tau$ is a complete Heyting algebra with Heyting implication $U \Rightarrow V = \bigcup\{W \in \tau : U \cap W \subseteq V\}$.

Proposition. Every metric space $(M, d)$ carries a canonical topology whose open sets are the unions of open balls $B(x, \varepsilon) = \{y \in M : d(x, y) < \varepsilon\}$.

Proposition. The category $\mathbf{Top}$ of topological spaces and continuous maps admits arbitrary products and coproducts, as well as subspace and quotient constructions.

Examples.

1. Standard topology on $\mathbb{R}$. Generated by open intervals $(a, b)$. Equivalently, the metric topology for $d(x, y) = |x - y|$.
2. Discrete topology. Every subset of $X$ is open. Every function out of a discrete space is continuous.
3. Indiscrete topology. Only $\emptyset$ and $X$ are open. Every function into an indiscrete space is continuous.
4. Cofinite topology. The open sets are $\emptyset$ and all subsets whose complement is finite.

Remark (categorical and algebraic perspectives). Viewing $(X, \tau)$ as a category whose objects are open sets and whose morphisms are inclusions is the starting point of sheaf theory: a presheaf assigns data to each open set, and a sheaf is a presheaf whose local data glues uniquely. The frame description of $\tau$ generalizes to locales (pointless topology) and to the closure operators studied in order theory.