Definition talk:T3 Space/Definition 3

Mistake in Definition and Display Error

In the set $\set {N_H: \relcomp S H \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$ is the complement $\relcomp S H$ worng and $H$ should be swaped with $N_H$. But then there is a display error $\relcomp S N_H$. The Index should be in the brackets. --SimonK (talk) 12:20, 20 June 2024 (UTC)

$\relcomp S H \in \tau$ means that $\relcomp S H$ is open, meaning $H$ is closed, which we want to specify.

Then we are stating that $N_H$ is the neighbourhood of that closed $H$.

Hence the symbology states: for all closed sets $H$ of $T$, $H$ is the intersection of all the sets $N_H$ such that there exists an open set $V$ "between" $H$ and its $N_H$, that is, the intersection of all its neighbourhoods.

Which is exactly what the definition of $T_3$ space states. --prime mover (talk) 23:44, 20 June 2024 (UTC)

I am not sure that these definitions are equivalent if we take only closed neighbourhoods or all neighbourhoods. But for the consistency of $\mathsf{Pr} \infty \mathsf{fWiki}$, this definition should be the same at every point. So in the Proof 'Eqivelences of the 3 T3 Definitions' this definition should be used and at every point the textual description should be changed from:

• $T = \struct {S, \tau}$ is $T_3$ if and only if each of its closed sets is the intersection of its closed neighborhoods:
• $T = \struct {S, \tau}$ is $T_3$ if and only if each of its closed sets is the intersection of its neighborhoods:

But in the book cited, the textual definition is as follows, "$T_3$ spaces may be characterized either by the fact that each open set contains a closed neighborhood around each of its points, or by the property that each closed set is the intersection of its closed neighborhoods." 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms

It would be great if we could just change it:

• From: $\bigcap \set {N_H: \relcomp S H \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$
• To: $\bigcap \set {N_H: \relcomp S {N_H} \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$ --SimonK (talk) 13:12, 22 June 2024 (UTC)

Apologies, you mentally derailed me with that "But then there is a display error $\relcomp S N_H$." Couldn't grok what you stated. Sorted now thx --prime mover (talk) 14:18, 22 June 2024 (UTC)