Open Set

Context

There are two definitions for open sets. The first is based on a metric.

Statements

Definition. (@Eck23, Definition 1.1) Let $(X,d)$ be a metric space. A subset of $S$ is said to be open in $X$ if, for every $s\in S$, there exists $\epsilon >0$ such that the open ball $B_{<\epsilon }(x)$ satisfies $B_{<\epsilon }(s)\subseteq S$.

Link to original

The second, is based on the concept of topological space.

Statements

Definition (@Tu11, pp. 318). Let $(X,\tau )$ be a topological space, a set $S\subset \tau$ is said to be an open set of $X$.

Link to original

The following proposition shows that the metric-based definition implies the second.

Statements

Proposition. Let $(X,d)$ be a metric space and $S\subseteq X$ be a metric open set. Then, $S$ is also a topological open set.

Proof. From the Topological Space Induced by the Metric Space, metric open sets form a topology in $X$.

Remark

A topological open set does not necessary implies a metric open sets.