Topology/Subspaces

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

Definition

Let $(X,𝒯)$ 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 $U\in 𝒯$ such that $U_1=U\cap X_1$

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.

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