Topology/Subspaces

Put simply, a subspace is a subset of a topological space. Subspaces have powerful applications in topology.

Definition

Let $(X,\tau )$ be a topological space, and let X₁ be a subset of X. Define the open sets as follows:

A set $U_1\subseteq X_1$ is open in $X_1$ there exists a a set U∈τ such that U₁=U∩X₁.

An important idea to note from the above definitions are:

A set not being open or closed does not prevent it from being open or closed within a subspace. For example, $(0,1)$ as a subspace of itself is both open and closed.