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


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.

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