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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Functions and Sets

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We conclude this section with a connection between subsets and functions. A bit of notation first. If f is a function from a set X to a set Y, and if A is a subset of X and B is a subset of Y, we define f(A) and fβˆ’1(B) as
f(A)={f(a)∣a∈C},
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and
fβˆ’1(B)={a∈A∣f(a)∈B}.
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We call f(A) the image of the set A under f and fβˆ’1(B) is the preimage of the set B under f. Note that fβˆ’1(B) is defined for any function, not just invertible functions. So it is important to recognize that the use of the notation fβˆ’1(B) does not imply that f is invertible.
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When we work with continuous functions in later sections, we will need to understand how a function behaves with respect to subsets. One result is in the following lemma.
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Proof.

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Let f:X→Y be a function and let {Aα} be a collection of subsets of X for α in some indexing set I. To prove part 1, we demonstrate the containment in both directions.
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Let b∈f(β‹ƒΞ±βˆˆIAΞ±). Then b=f(a) for some aβˆˆβ‹ƒΞ±βˆˆIAΞ±. It follows that a∈Aρ for some ρ∈I. Thus, b∈f(Aρ)βŠ†β‹ƒΞ±βˆˆIf(AΞ±). We conclude that f(β‹ƒΞ±βˆˆIAΞ±)βŠ†β‹ƒΞ±βˆˆIf(AΞ±).
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Now let bβˆˆβ‹ƒΞ±βˆˆIf(AΞ±). Then b∈f(Aρ) for some ρ∈I. Since AΟβŠ†β‹ƒΞ±βˆˆIAΞ±, it follows that b∈f(β‹ƒΞ±βˆˆIAΞ±). Thus, β‹ƒΞ±βˆˆIf(AΞ±)βŠ†f(β‹ƒΞ±βˆˆIAΞ±). The two containments prove part 1.
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For part 2, we again demonstrate the containments in both directions. Let a∈fβˆ’1(β‹ƒΞ²βˆˆJBΞ²). Then f(a)βˆˆβ‹ƒΞ²βˆˆJBΞ². So there exists μ∈J such that f(a)∈BΞΌ. This implies that a∈fβˆ’1(BΞΌ)βŠ†β‹ƒΞ²βˆˆJfβˆ’1(BΞ²). We conclude that fβˆ’1(β‹ƒΞ²βˆˆJBΞ²)βŠ†β‹ƒΞ²βˆˆJfβˆ’1(BΞ²).
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For the reverse containment, let aβˆˆβ‹ƒΞ²βˆˆJfβˆ’1(BΞ²). Then a∈fβˆ’1(BΞΌ) for some μ∈J. Thus, f(a)∈BΞΌβŠ†β‹ƒΞ²βˆˆJBΞ². So a∈fβˆ’1(β‹ƒΞ²βˆˆJBΞ²). Thus, β‹ƒΞ²βˆˆJfβˆ’1(BΞ²)βŠ†fβˆ’1(β‹ƒΞ²βˆˆJBΞ²). The two containments verify part 2.
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At this point it is reasonable to ask if Lemma 2.11 would still hold if we replace unions with intersections. We leave that question for Exercise 7.
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Another result is contained in the next activity.
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Activity 2.8.

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Let X, Y, and Z be sets, and let f:Xβ†’Y and g:Yβ†’Z be functions. Let C be a subset of Z. There is a relationship between (g∘f)βˆ’1(C) and fβˆ’1(gβˆ’1(C)). Find and prove this relationship.
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