Topology/Subspaces

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

Definition

Let $( X,\mathcal{T})$ 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 X₁ if there exists a a set $U\in\mathcal{T}$ such that $U_1=U\bigcap X_1$

An important idea to note from the above definitions is that 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.