Equivalence of Definitions of Irreducible Space/3 iff 6

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Theorem

The following definitions of the concept of Irreducible Space are equivalent:

Definition 3

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

$\forall U, V \in \tau: U, V \ne \varnothing \implies U \cap V \ne \varnothing$

Definition 6

A topological space $T = \left({S, \tau}\right)$ is irreducible if and only if the closure of every non-empty open set is the whole space:

$\forall U \in \tau: U^- = S$


Proof

Let $T = \left({S, \tau}\right)$ be an irreducible space by Definition 3.

That is, let every non-empty open set in $T$ have a non-empty intersection with every other non-empty open set.

Let $U \in \tau$ be open in $T$ such that $U \ne \varnothing$.

If $U = S$ then $U^- = S$ trivially.


So, let $U \ne S$.

Let $x \in S$ such that $x \ne U$.

Let $V$ be any non-empty open set of $T$ such that $x \in V$.

By hypothesis, $U \cap V \ne \varnothing$.

Therefore $V$ contains some point of $U$ distinct from $x$.

$V$ is arbitrary, so it follows that every open set $V$ of $T$ such that $x \in V$ contains some point of $U$ distinct from $x$.

That is, $x$ is a limit point of $U$.

So every point of $T$ is a limit point of $U$.

Thus, by definition, every point of $T$ is in the closure of $U$.


The argument reverses directly.

$\blacksquare$


Sources