Locally Complete Metric Space

Question

It is undoubtedly the concept of "Complete Metric Space" is well known for all the users.

The Locally Complete Metric Space is a metric space where each point has a neighbourhood (which is closed) is a complete metric space.

It is so obvious that every complete metric space is locally complete metric space.

I am looking for an example of locally complete metric space that is not complete metric space.

Answer 1

Example 1: $\lbrace \dfrac1n: n \in \Bbb N\rbrace$ with the usual metric.

Example 2: Any open but non-closed subset $U$ of a complete metric space. Like open intervals of finite length in $\Bbb R$.

How to bring these examples together? I think the following characterisation is valid: Let $A$ be a subset of a complete metric space. Then $A$ with the restricted metric is locally complete but not complete if and only if:

1. $A$ is not closed
2. $cl(cl(A) \setminus A) \cap A = \emptyset$, or equivalently, no limit point of $cl(A)\setminus A$ lies in $A$, or equivalently, for every $a\in A$, $\inf \lbrace d(a, x): x\in cl(A)\setminus A\rbrace > 0$.

Since every metric space embeds into its completion, this gives a full (well, up to describing completions and checking these properties ...) characterisation of the spaces you ask for.