Open and Closed Sets in Subspaces

Section Open and Closed Sets in Subspaces

We saw in our preview activity that a subspace does not necessarily inherit the properties of the larger space. For example, a subset of a subspace might be open in the subspace, but not open in the larger space. However, there is a connection between the open subsets in a subspace and the open sets in the larger space.

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Theorem 11.2.

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Let

(X, d)

be a metric space and

A

a subset of

X .

A subset

O_{A}

A

is open in

A

if and only if there is an open set

O_{X}

X

so that

O_{A} = O_{X} \cap A .

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Proof.

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Let

(X, d)

be a metric space and

A

a subset of

X .

Let

O_{A}

be an open subset of

A .

So for each

a \in A

there is a

δ_{a} > 0

so that

B_{A} (a, δ_{a}) \subseteq O_{A},

where

B_{A} (a, δ_{a})

is the open ball in

A

centered at

a

of radius

δ_{a} .

Then,

O_{A} = ⋃_{a \in O_{A}} B_{A} (a, δ_{a}) .

Now let

B_{X} (a, δ_{a})

be the open ball in

X

centered at

a

of radius

δ_{a},

and let

O_{X} = ⋃_{a \in O_{A}} B_{X} (a, δ_{a}) .

Note that

B_{A} (a, δ_{a}) = B_{X} (a, δ_{a}) \cap A .

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As a union of open balls in

X,

the set

O_{X}

is open in

X .

Now

\begin{aligned} O_{X} \cap A = (⋃_{a \in O_{A}} B_{X} (a, δ_{a})) \cap A & = ⋃_{a \in O_{A}} (B_{X} (a, δ_{a}) \cap A) \\ = ⋃_{a \in O_{A}} B_{A} (a, δ_{a}) = O_{A} . \end{aligned}

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So there is an open set

O_{X}

X

such that

O_{A} = O_{X} \cap A .

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For the reverse implication, see the following activity.

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Activity 11.2.

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Let

(X, d)

be a metric space and

A

a subset of

X .

Suppose that

O_{A} = A \cap O_{X}

for some set

O_{X}

that is open in

X .

In this activity we will prove that

O_{A}

is an open subset of

A .

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(a)

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Let

a \in O_{A} .

Explain why there must exist a

δ > 0

such that

B_{X} (a, δ),

the open ball in

X

of radius

δ

around

a

X,

is a subset of

O_{X} .

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(b)

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What would be a natural candidate for an open ball in

A

centered at

a

that is contained in

A ?

Prove your conjecture.

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(c)

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What conclusion can we draw?

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We might now wonder about closed sets in a subspace. If

X

is a metric space and

A

is a subspace, then by definition a subset

C_{A}

A

is closed if and only if

C_{A} = A ∖ O_{A}

for some set

O_{A}

that is open in

A .

The analogy of Theorem 11.2 is true for closed sets in subspaces.

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Theorem 11.3.

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Let

(X, d)

be a metric space and

A

a subset of

X .

A subset

C_{A}

A

is closed in

A

if and only if there is a closed set

C_{X}

X

so that

C_{A} = C_{X} \cap A .

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The proof is left to Exercise 4.

Topology: An Inquiry-Based Approach

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Section Open and Closed Sets in Subspaces

Theorem 11.2.

Proof.

Activity 11.2.

(a)

(b)

(c)

Theorem 11.3.