Definition:Minimal Hausdorff Topology

Definition

Examples of neighborhood bases of $a$ and $-a$

Let $\omega$ be the first transfinite ordinal.

Let $\struct {A, \tau_1}$ denote the totally ordered set defined as:

$A := \set {1, 2, 3, \ldots, \omega, \ldots, -3, -2, -1}$

with the interval topology $\tau_1$.

Let $\struct {\Z_{>0}, \tau_2}$ denote the (strictly) positive integers with the discrete topology $\tau_2$.

Let $S$ be defined as:

$S = A \times \Z_{>0} \cup \set {a, -a}$

where $a$ and $-a$ are new elements of $S$.

Let $\tau$ be the topology defined on $S$ as:

the product topology on $\struct {A, \tau_1} \times \struct {\Z_{>0}, \tau_2}$

together with neighborhood bases of $a$ and $-a$ defined as:

$\ds \map { {M_n}^+} a$

$:=$

$\ds \set a \cup \set {\tuple {i, j}: i < \omega, j > n}$

$\ds \map { {M_n}^-} {-a}$

$:=$

$\ds \set {-a} \cup \set {\tuple {i, j}: i > \omega, j > n}$

$\tau$ is referred to as the minimal Hausdorff topology.

The topological space $T = \struct {S, \tau}$ is referred to as the minimal Hausdorff space.

This entry was named for Felix Hausdorff.

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $100$. Minimal Hausdorff Topology