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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Components

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As Activity 18.6 demonstrates, spaces like A=(1,2)βˆͺ(3,4) are not connected. Even so, A is made of two connected subsets (1,2) and (3,4). These connected subsets are called components.
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Definition 18.9.

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A subspace C of a topological space X is a component (or connected component) of X if C is connected and there is no larger connected subspace of X that contains C.
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As an example, if X=(1,2)βˆͺ[4,10)βˆͺ{βˆ’1,15}, then the components of X are (1,2), [4,10), {βˆ’1} and {15}. As the next activity shows, we can always partition a topological space into a disjoint union of compenents.
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Activity 18.7.

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Let (X,Ο„) be a nonempty topological space.
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(b)

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Suppose that X is a topological space and {AΞ±} for Ξ± in some indexing set I is a collection of connected subsets of X. Let A=β‹ƒΞ±βˆˆIAΞ±. Suppose that β‹‚Ξ±βˆˆIAΞ±β‰ βˆ…. Show that A is a connected subset of X.
Hint.
Assume a separation and use Lemma 18.7.
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(c)

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Part (a) shows that every element in x belongs to some connected subset of X. So we can write X as a union of connected subsets. But there is probably overlap. To remove the overlap, we define the following relation ∼ on X: For x and y in X, x∼y if x and y are contained in the same connected subset of X. As with any relation, we ask if ∼ is an equivalence relation.
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Activity 18.7 shows that the relation ∼ is an equivalence relation, and so partitions the underlying topological space X into disjoint sets. If x∈X, then the equivalence class of x is a connected subset of X. There can be no larger connected subset of X that contains x, since equivalence classes are disjoint or the same. So the equivalence classes are exactly the connected components of X. The components of a topological space X satisfy several properties.
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    Each a∈X is an element of exactly one connected component Ca of X.
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    A component Ca contains all connected subsets of X that contain a. Thus, Ca is the union of all connected subsets of X that contain a.
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    If a and b are in X, then either Ca=Cb or Ca∩Cb=βˆ….
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    Every connected subset of X is a subset of a connected component.
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    Every connected component of X is a closed subset of X.
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    The space X is connected if and only if X has exactly one connected component.
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