**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 sub*space*.

**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 *U* ∩ *Y*, *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* : *Z* → *X*, 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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