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 $$X_1$$ be a subset of $$X$$. Define the open sets as follows:

A set $$U_1 \subseteq X_1$$ is open in $$X_1$$ 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.