The Subspace Topology

Section The Subspace Topology

In our preview activity, we saw that the intersection of the open sets in a topological space

X

with any nonempty subset

A

X

forms a topology for

A .

We then have

A

as a subspace of

X .

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The topology

τ_{A}

in Definition 15.2 is called the subspace topology, the induced topology, or the relative topology. In our preview activity we saw that sets that are open in a subspace

A

of a topological space

X

need not be open in

X .

So we call the sets in

τ_{A}

relatively open.

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Once we have defined relatively open sets, we can then consider how to define relatively closed sets.

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Activity 15.2.

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Let

(X, τ)

be a topological space, and let

A

be a subset of

X .

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(a)

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Recall that a subset of a topological space is closed if its complement is open. Given that

(A, τ_{A})

is a topological space, how is a closed set in

A

defined? Such a set will be called relatively closed.

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(b)

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Recall that a subset

U

A

is relatively open if and only if

U = A \cap O

for some open subset of

X .

With this in mind, how might we expect a relatively closed set in

A

to be related to a closed set in

X ?

State and prove a theorem for this result.

Topology: An Inquiry-Based Approach

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Section The Subspace Topology

Activity 15.2.

(a)

(b)