Definition:Component (Topology)/Definition 3
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let the relation $\sim $ be defined on $T$ as follows:
- $x \sim y$ if and only if $x$ and $y$ are connected in $T$.
That is, if and only if there exists a connected set of $T$ that contains both $x$ and $y$.
The component of $T$ containing $x$ is defined as:
- the maximal connected set of $T$ that contains $x$.
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.5$: Components
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness