A First Course in Topology by Robert A. Conover

Author:Robert A. Conover
Language: eng
Format: epub, azw3
Publisher: Dover Publications
Published: 2014-04-10T04:00:00+00:00

3. COMPONENTS AND LOCAL CONNECTIVITY

As we have seen, (0, 1) ∪ (1, 2) is not a connected subset of R with its usual topology. However, both (0, 1) and (1,2) are connected, and, furthermore, for each point in (0,1), (0,1) is the largest connected subset of (0,1) ∪ (1,2) containing the point. Similarly, (1, 2) is the largest connected subset of (0, 1) u (1, 2) containing each of its points. In general,

3.1. Definition. Let x be a topological space and let x ∈ X. The largest connected subset of X containing x is called the component of X that contains x, and is denoted by Cx.

Thus for x ∈ X, Cx is a connected subset of X containing x such that if C is a connected subset of X which also contains x, then C ⊂ Cx. When dealing with components it is often easier to use the following theorem instead of the definition.

3.2. Theorem. Let X be a topological space and let x ∈ X. Then

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