Existence of Minimal Hausdorff Space which is not Compact
Theorem
Let $S$ be a set.
Let $\tau$ be the minimal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a Hausdorff space.
Then it is not necessarily the case that $\struct {S, \tau}$ is compact.
Proof
Let $T = \struct {S, \tau}$ be the canonical minimal Hausdorff non-compact space.
This space has been so named on $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to allow reference to it without needing to describe it whenever it is mentioned.
By Canonical Minimal Hausdorff Non-Compact Space is Minimal Hausdorff, $\tau$ is the minimal subset of the power set $\powerset S$ such that $T$ is a Hausdorff space.
By Canonical Minimal Hausdorff Non-Compact Space is not Compact, $T$ is not a compact topological space.
Hence the result.
$\blacksquare$
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: Compactness Properties and the $T_i$ Axioms