Definition:Telophase Topology/Mistake

Source Work

 * Part $\text {II}$: Counterexamples
 * Section $73$: Telophase Topology
 * 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 of a local neighborhood basis.

They define a local basis, but not a neighborhood basis, for which has taken the definition from  by  (1962).

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.

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

Hence it would be appropriate for in this instance to say local basis where they currently say local neighborhood basis.

Also see

 * Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected