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.
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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