Particular Point Space is Irreducible

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.


Then $T$ is irreducible.


Proof 1

By definition, $T = \left({S, \tau}\right)$ is irreducible if and only if every two non-empty open sets of $T$ have non-empty intersection.

Let $U_1$ and $U_2$ be non-empty open sets of $T$.

By definition of particular point space, $p \in U_1$ and $p \in U_2$.

Thus:

$p \in U_1 \cap U_2$

and so:

$U_1 \cap U_2 \ne \varnothing$

Hence the result.

$\blacksquare$


Proof 2

Follows directly from:

Particular Point Topology is Closed Extension Topology of Discrete Topology
Closed Extension Space is Irreducible

$\blacksquare$