Definition:Open Extension Topology

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $p$ be a new element which is not in $S$.

Let $S^*_p = S \cup \set p$.

Let $\tau^*_p$ be the set defined as:

$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$

That is, $\tau^*_{\bar p}$ is the set of all sets formed by taking all the open sets of $\tau$ and adding to them the set $S^*_p$.

Then:

$\tau^*_{\bar p}$ is the open extension topology of $\tau$

and:

$T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ is the open extension space of $T = \struct {S, \tau}$.

The open sets of $T^*_{\bar p}$ can be seen to be the same as the open sets of $T$, but with $S^*_p$ added.

Also see

• Open Extension Topology is Topology
• Definition:Closed Extension Topology

• Results about open extension topologies can be found here.

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: $8$