Double Pointed Real Number Space is Weakly Countably Compact

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Theorem

Let $T_R = \struct {\R, \tau_R}$ be the (uncountable) discrete space on the set of real numbers.

Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$.

Let $T = T_R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_R$ and $T_D$.


Then $T$ is weakly countably compact.


Proof

We have that $T$ is a partition topology, whose basis $\PP$ is defined as:

$\PP = \set {\set {\tuple {s, a}, \tuple {s, b} }: s \in \R}$

Let $A \subseteq \R \times D$ such that $A$ is infinite.

Let $x \in A$.

Let $U$ be the union of sets of the form $\set {\tuple {s, a}, \tuple {s, b} }$, and hence open in $T$.

Now if $x \in U$, it will be an element in some $\set {\tuple {s, a}, \tuple {s, b} }$.

So there will exist $y \in U$ which will also be an element in that $\set {\tuple {s, a}, \tuple {s, b} }$.

So, by definition, $x$ is a limit point of $A$.

So, by definition, $T$ is weakly countably compact.

$\blacksquare$


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