Topology: An Inquiry-Based Approach

A subspace of a topological space is any nonempty subset of the topological space endowed with the subspace topology.

An open subset in the subspace topology for a subset $A$ of a topological space $X$ is any set of the form $O\cap A\text{,}$ where $O$ is an open set in $X\text{.}$

The relatively open sets are the open sets in a subspace topology. The relatively closed sets are complements of the relatively open sets in a subspace topology. That is, a relatively closed set in the subspace $A$ of a topological space $X$ are the sets of the form $A\cap C\text{,}$ where $C$ is a closed set in $X\text{.}$

The topological space $\mathbb{R}$ with the standard topology is homeomorphic to any open interval as well as open intervals of the form $(a,\mathrm{\infty })$ or $(-\mathrm{\infty },b)$ for any real numbers $a$ and $b\text{.}$