Open Extension Space is Connected
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a connected space.
Proof
The only open set of $T$ which contains $p$ is $S^*_p$.
Therefore it is impossible to set up a separation of $T$, as $S^*_p$ will always need to be an element of such a separation.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $16$. Open Extension Topology: $9$