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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Continuity and Closed Sets

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Recall that we showed that a function f from a metric space (X,dX) to a metric space (Y,dY) is continuous if and only if f−1(O) is open for every open set O in Y. We might conjecture that a similar result holds for closed sets. Since closed sets are complements of open sets, to make this connection we will want to know how X∖f−1(B) is related to f−1(Y∖B) for B⊂Y.
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Activity 10.4.

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Let f be a function f from a metric space (X,dX) to a metric space (Y,dY), and let B be a subset of Y.
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(a)

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Let x∈X∖f−1(B).
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(i)
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What does this tell us about f(x)?
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(ii)
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What can we conclude about the relationship between X∖f−1(B) and f−1(Y∖B)?
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(b)

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Let x∈f−1(Y∖B).
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(i)
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What does this tell us about f(x)?
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(ii)
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What can we conclude about the relationship between X∖f−1(B) and f−1(Y∖B)?
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(c)

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What is the relationship between X∖f−1(B) and f−1(Y∖B)?
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Now we can consider the issue of continuity and closed sets.
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Activity 10.5.

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Let f be a function from a metric space (X,dX) to a metric space (Y,dY).
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(a)

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Assume that f is continuous and that C is a closed set in Y. How does the result of Activity 10.4 tell us that f−1(C) is closed in X?
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(b)

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Now assume that f−1(C) is closed in X whenever C is closed in Y. How does the result of Activity 10.4 tell us that f is a continuous function?
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The result of Activity 10.5 is summarized in the following theorem.
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