# Topology/Subspaces

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

## Definition[edit]

Let be a topological space, and let be a subset of . Define the open sets as follows:

A set is **open** in if there exists a a set such that

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, as a subspace of itself is both open and closed.