Integer Reciprocal Space with Zero is not Extremally Disconnected
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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:
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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $2 \ \text{(d)}$