Particular Point Space is Irreducible
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Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is irreducible.
Proof 1
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.
$\blacksquare$
Proof 2
Follows directly from:
$\blacksquare$