Particular Point Space is Irreducible/Proof 1

Theorem

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

Then $T$ is irreducible.

Proof

By definition, $T = \struct {S, \tau_p}$ 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:

$U_1 \cap U_2 \ne \O$

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $10$