Modified Fort Space is Scattered
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Theorem
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is scattered.
Proof
We have that a modified Fort space is $T_1$.
We also have that a dense-in-itself subset of a $T_1$ space is infinite.
But from Isolated Points in Subsets of Modified Fort Space, we have that any subset of $T$ with more than two points has at least one isolated point.
So any dense-in-itself subset of $T$ must have an isolated point.
Hence the result, by definition of scattered space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $27$. Modified Fort Space: $6$