Existence of Compact Space which Satisfies No Separation Axioms
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Theorem
There exists at least one example of a compact space for which none of the Tychonoff separation axioms are satisfied.
Proof
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Let $D = \struct {\set {0, 1}, \vartheta}$ be the indiscrete topology on two points.
Let $T \times D$ be the double pointed topology on $T$.
From Double Pointed Finite Complement Topology is Compact, $T \times D$ is compact.
From Double Pointed Finite Complement Topology fulfils no Separation Axioms, $T \times D$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space, $T_4$ space or $T_5$ space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness