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After studying several kinds of topological spaces (Like $L_p, C[0,1]$) etc., I thought it would be useful to me (and to others) if I tabulated some of them under 3 categories: Completeness(C), Separability(S) and Metrizability(M). Now this yields a total of 8 possibilities from spaces that are neither to spaces that are all 3. So far I have the following:

C,S,M : $\mathbb{R}^n$, $C[0,1]$, $L_p(\mu), 1\leq p < \infty$ where $\mu$ is the Lebesgue measure.

C,~S,M : $L_{\infty}(\mu)$

~C,S,M : $D[0,1]$ space under the Skorokhod's metric. However, under Billingsley's Metric (ref. Richard Bass "Stochastic Processes"), it is C,S,M.

~C,S,~M : The Lower Limit topology of $\mathbb{R}$.

Since completeness is a property of a metric space, you can never have (C,~M). That eliminates two cases.

So that leaves me with (~C,~S,M) and (~C,~S,~M) for which I have no examples as of yet. I require at least 1 example in each to complete my list.

Question: Kindly help me find the remaining examples. References are welcome.

My searches: I searched for non complete non separable non metrizable spaces, but that didn't add to the table. Google searches were usually putting the "non" in front of separable and not anywhere else.

Purpose: I offer you my word that this is not a homework. Although it is interesting in it's own merit.

Thanks to everyone who helps make the table.

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You can have complete and non-metrisable if you allow yourself to study uniform spaces such as topological groups. I believe that the following is a usable example: Consider the ordered field $K := \mathbb Q(\epsilon)$ where $\epsilon$ is a positive infinitesimal, i.e. for all positive rationals $q$ we have $0 < \epsilon < q$ and $\epsilon^n \to 0$ as $n\to\infty$. The construction using Dedekind cuts of $K$ would yield a complete uniform space which is not metrisable. – kahen May 12 '13 at 14:36
Probably the simplest such topological space is that of ordinal numbers. I'm pretty sure about it not being separable, and pretty confident it neither is metrizable. – AndreasT May 12 '13 at 15:09
up vote 2 down vote accepted

The book Counterexamples in Topology by Steen and Seebach would interest you. In the back it has a big table of hundreds of spaces and their properties. From that table I found a number of examples to (~C,~S,~M), for example the countable complement topology (there is a proof on M.SE: Properties of the countable complement topology on $\mathbb{R}$. ). That table does not list completeness, so I couldn't at a glance find an example of (~C,~S,M), but if one looks up all the spaces which are (~S,M) in the book, one might find one.

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Incomplete inseparable metric space is easy: Start with a complete inseparable metric space (OP already knows one) and take a sufficiently big dense subspace. – kahen May 12 '13 at 15:13

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