In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any set and p ∈ X. The collection
\( {\displaystyle T=\{S\subseteq X:p\notin S{\text{ or }}S=X\}} \)
of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named:
If X has two points, it is called the Sierpiński space. This case is somewhat special and is handled separately.
If X is finite (with at least 3 points), the topology on X is called the finite excluded point topology
If X is countably infinite, the topology on X is called the countable excluded point topology
If X is uncountable, the topology on X is called the uncountable excluded point topology
A generalization is the open extension topology; if \( {\displaystyle X\setminus \{p\}} \) has the discrete topology, then the open extension topology on \( {\displaystyle (X\setminus \{p\})\cup \{p\}} \) is the excluded point topology.
This topology is used to provide interesting examples and counterexamples. A space with the excluded point topology is connected, since the only open set containing the excluded point is X itself and hence X cannot be written as disjoint union of two proper open subsets.
See also
Alexandrov topology
Finite topological space
Fort space
Particular point topology
References
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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