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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Introduction

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Let (X,d) be a metric space, and let A be a subset of X. We can make A into a metric space itself in a very straightforward manner. When we do so, we say that A is a subspace of X.
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Preview Activity 11.1.

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Let (X,d) be a metric space, and let A be a subset of X. To make the subset A into a metric space, we need to define a metric on A. For us to consider A as a subspace of X, we want the metric on A to agree with the metric on X. So we define d′:A×A→R by
d′(a1,a2)=d(a1,a2)
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for all a1,a2∈A. Note that d and d′ are different functions because they have different domains.
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(a)

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Show that d′ is a metric on A. Since d′ is a metric on A it follows that (A,d′) is a metric space. The metric d′ is the restriction of d to A×A and can also be denoted by dA.
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Definition 11.1.
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Let (X,d) be a metric space. A subspace of (X,d) is a subset A of X together with the metric dA from A×A to R defined by
dA(a1,a2)=d(a1,a2)
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for all a1,a2∈A.
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We might wonder what, if any, properties of the space X are inherited by a subspace.
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(b)

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Let (X,d)=(R,dE) and let A=[0,1]. Let 0<a<1. Is the set [0,a) open in X? Is the set [0,a) open in A? Explain.
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(c)

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Let (X,d)=(R,dE) and let A=Z. What are the open subsets of A? Explain.
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(d)

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Let (X,d)=(R2,dE), let A={(x,0)∣x∈R} (the x-axis in R2), and let Z={(x,0)∣0<x<1}. Note that Z⊂A⊂X and we can consider Z as a subspace of A and X, and A as a subspace of X.
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(i)
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Explain why A is a closed subset of X.
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(ii)
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Explain why Z is an open subset of A.
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(iii)
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Is Z an open subset of X? Explain
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