# Integer Reciprocal Space with Zero is not Extremally Disconnected

Jump to navigation Jump to search

## Theorem

Let $A \subseteq \R$ be the set of all points on $\R$ defined as:

$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$

Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.

Then $A$ is not extremally disconnected.

## Proof

$\struct {A, \tau_d}$ is a metric space.

We have:

Extremally Disconnected Metric Space is Discrete

We also have:

Topological Space is Discrete iff All Points are Isolated
Zero is Limit Point of Integer Reciprocal Space

From definition of limit points:

$0$ is not an isolated point of $A$

Hence integer reciprocal space with zero is not the discrete space, and the result follows.

$\blacksquare$