Clustering of Cofinally Cauchy nets

Question

If $(X,d)$ is a metric space in which every Cofinally Cauchy sequence clusters. Does this imply every Cofinally Cauchy net clusters in the space?

Answer 1

Yes, it does.

In the terminology of Gerald Beer, Between compactness and completeness, Topology and its Applications $155$ ($2008$), $503$-$514$, a sequence $\langle x_n:n\in\Bbb N\rangle$ in a metric space $\langle X,d\rangle$ is cofinally Cauchy if for each $\epsilon>0$ there is an infinite $N_\epsilon\subseteq\Bbb N$ such that $d(x_k,x_\ell)<\epsilon$ whenever $k,\ell\in N_\epsilon$. $X$ is cofinally complete if every cofinally Cauchy sequence in $X$ has a cluster point.

Beer proves the following theorem.

Theorem $\bf3.2$. Let $\langle X,d\rangle$ be a metric space. The following are equivalent:

1. $X$ is cofinally complete.
2. Whenever $\langle x_n:n\in\Bbb N\rangle$ is a sequence in $X$ with $\lim\limits_{n\to\infty}\nu(x_n)=0$, then $\langle x_n:n\in\Bbb N\rangle$ has a cluster point.
3. Either $X$ is uniformly locally compact, or $\operatorname{nlc}(X)$ is non-empty and compact, and $\left\langle\left\{x:\nu(x)\le\frac1n\right\}:n\in\Bbb Z^+\right\rangle$ convertes to $\operatorname{nlc}(X)$ in Hausdorff distance.
4. $\operatorname{nlc}(X)$ is compact and for all $\epsilon>0$, $X\setminus\big(\operatorname{nlc}(X)\big)^\epsilon$ is uniformly locally compact in its relative topology.

Here $A^\epsilon=\bigcup_{x\in A}B(x,\epsilon)$ for $A\subseteq X$, and $\nu(x)$ for $x\in X$ is defined as follows. If $x$ has a compact nbhd, then $$\nu(x)=\sup\{\epsilon>0:\overline{B}(x,\epsilon)\text{ is compect}\}\;,$$ where $\overline{B}(x,\epsilon)=\{y\in X:d(x,y)\le\epsilon\}$. If $x$ has no compact nbhd, then $\nu(x)=0$. Finally, $\operatorname{nlc}(X)=\{x\in X:\nu(x)=0\}$ is the set of points of non-local compactness of $X$. The proof of the theorem is followed by several remarks. I quote the relevant one:

Condition $(4)$ is a variant of a condition presented by Hohti [$12$, Thm $2.1.1$] in his thesis that characterizes uniform paracompactness in metric spaces as defined by Rice [$22$]. Since uniform paracompactness is equivalent to net cofinal completeness in a general uniform space [$25,14$], it follows that sequential cofinal completeness and the formally stronger net cofinal completeness coincide in the context of metric spaces. This is not always the case in uniform spaces. In fact, Burdick [$7$] has given an example showing that cofinal completeness for all transfinite sequences may be properly weaker than net cofinal completeness.

The references whose authors are named are:

• A. Hohti, On uniform paracompactness, Ann. Acad. Sci. Fenn. Ser. A Math. Dissertationes $36$ ($1981$), $1$-$46$.
• M. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc. $62$ ($1977$), $359$-$362$.
• B. Burdick, On linear cofinal completeness, Topology Proc. $25$ ($2000$), $435$-$455$.

[$25$] is J. Smith, Review of ‘A note on uniform paracompactness’ by Michael D. Rice, Math. Rev. $55$ ($1978$), #$9036$, and [$14$] is N. Howes, Modern Analysis and Topology, Springer, New York, $1995$.