Open Set

Let $(X, \tau)$ be a topological space.

Definition. A subset $U \subseteq X$ is an open set if $U \in \tau$.

Open sets are the primitive notion of a topology: they are not defined by an intrinsic property of $U$ but declared by membership in $\tau$. The axioms on $\tau$ require exactly:

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).

Proposition. In a metric space $(X, d)$, a subset $U \subseteq X$ is open in the metric topology if and only if for every $x \in U$ there exists $\varepsilon > 0$ such that $B(x, \varepsilon) \subseteq U$.

Proof sketch. The open balls $\{B(x, \varepsilon) : x \in X,\, \varepsilon > 0\}$ form a basis for the metric topology. A set is open if and only if it is a union of basis elements, which is equivalent to the stated condition. $\square$

Proposition. The collection $\tau$, ordered by inclusion, is a complete lattice that is a frame: join is union, meet is intersection, and finite meets distribute over arbitrary joins. Consequently $\tau$ is a complete Heyting algebra with Heyting implication $U \Rightarrow V = \bigcup \{ W \in \tau : U \cap W \subseteq V \}$.

Proof sketch. Arbitrary joins (unions) and binary meets (intersections) exist by axiom. Distributivity holds because $U \cap \bigcup_i V_i = \bigcup_i (U \cap V_i)$. A complete lattice in which finite meets distribute over arbitrary joins is a frame, and every frame is a complete Heyting algebra. $\square$

Examples.

1. Discrete topology. Every subset of $X$ is open, so $\tau = \mathcal{P}(X)$.
2. Indiscrete topology. Only $\emptyset$ and $X$ are open, so $\tau = \{\emptyset, X\}$.
3. Standard topology on $\mathbb{R}$. The open sets are exactly the arbitrary unions of open intervals $(a, b)$.

Remark. The frame structure of $\tau$ makes every topological space a model of intuitionistic logic. In this setting, the operator $j = \mathrm{int} \circ \mathrm{cl}$ (interior of closure) is a Lawvere-Tierney topology.