# Definition:Telophase Topology/Mistake

## Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology* (2nd ed.):

- Part $\text {II}$: Counterexamples
- Section $73$: Telophase Topology

## Mistake

*Let $\struct {X, \tau}$ be the topological space formed by adding to the ordinary closed unit topology $\sqbrk {0, 1}$ another right end point, say $1^*$, with the sets $\paren {\alpha, 1} \cup \set {1^*}$ as a local neighborhood basis.*

## Correction

There is no actual definition in *Counterexamples in Topology, 2nd ed.* of a **local neighborhood basis**.

They define a **local basis**, but not a **neighborhood basis**, for which $\mathsf{Pr} \infty \mathsf{fWiki}$ has taken the definition from *Introduction to Topology, 3rd ed.* by Bert Mendelson (1975).

From Local Basis Generated from Neighborhood Basis, a local basis can be generated from a neighborhood basis.

In fact, from Local Basis is Neighborhood Basis, if $\BB$ is a local basis, then it is *a fortiori* a neighborhood basis.

When a topological space is first-countable, a local basis and a neighborhood basis are the same thing.

A specific link is needed here.In particular: Neighborhood Basis in First_Countable Space is Local BasisYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by searching for it, and adding it here.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{LinkWanted}}` from the code. |

We have the result Telophase Topology is First-Countable, so that condition is fulfilled in this case.

Hence it would be appropriate for *Counterexamples in Topology, 2nd ed.* in this instance to say **local basis** where they currently say **local neighborhood basis**.

## Also see

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous): Part $\text {II}$: Counterexamples: $73$. Telophase Topology