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4.9 Subspaces

A subspace Y of a topological space X is just a subset of X. However, one needs to impose a topology on Y to make it a subspace.

Definition 4.9.1 (Subspace topology) Let X be a topological space, and Y be a subset of X. The subspace topology on Y has as opens the subsets of Y of the form UY, U open in X.

One then says that Y, with the subspace topology, is a subspace of X.

There is a slightly more general definition.

Definition 4.9.2 (Induced topology) Given any map f : ZX, where Z is a set and X is a topological space, the topology on Z induced from that of X by f is the coarsest topology on Z that makes f continuous.

The induced topology is simple: take as opens in Z exactly those subsets of the form f−1(U), U open in

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