Topology: An Inquiry-Based Approach

A subset $O$ of a metric space $(X,d)$ is an open set if $O$ is a neighborhood of each of its points. Alternatively, $O$ is open if $O$ is a union of open balls.

A point $a$ in a subset $A$ of a metric space $(X,d)$ is an interior point of $A$ if $A$ is a neighborhood of $a\text{.}$ A set $O$ is open if every point of $O$ is an interior point of $O\text{.}$

The interior of a set is the set of all interior points of the set. The interior of a set $A$ in a metric space $X$ is the largest open subset of $X$ contained in $A\text{.}$ A set is open if and only if the set is equal to its interior.

A function $f$ from a metric space $X$ to a metric space $Y$ is continuous if $f^{-1}(O)$ is open in $X$ whenever $O$ is open in $Y\text{.}$

Any union of open sets is open, while any finite intersection of open sets is open.