Metric and Topological Spaces

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## The subspace topology

We now consider some ways of getting new topologies from old ones.

Definition
If A is a subset of a topological space (X, X), we define the subspace topology A on A by:
B A if B = A C for some C X .

Examples

1. Restricting the metric on a metric space to a subset gives this topology.
For example, On X = [0, 1] with the usual metric inherited from R, the open sets are the intersection of [0, 1] with open sets of R.
So, for instance, [1, 1/4) = (-1, 1/4) [0, 1] and so is an open subset of the subspace X.

Remark
Note that as in this example, sets which are open in the subspace are not necessarily open in the "big space".

2. The subspace topology on Z R (with its usual topology/metric) is the discrete topology.

3. The subspace topology on the x-axis as a subset of R2 (with its usual topology) is the usual topology on R.

Remark
If we take the inclusion map i: A X then the subspace topology is the weakest topology (fewest open sets) on A in which this map is continuous.
Proof
If B X is open then i-1(B) = A B.

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JOC February 2004