Topological embeddings of non-compact, complete metric spaces

Question

Given a completely metrizable space, say that it has property X if it can be embedded in some metric space such that its image is not closed. For example, the real line R can be embedded, topologically, in itself as (0,1) which is not closed. A compact space such as S^1, however, clearly cannot be embedded in any metric space in this way.

1. Is it true that a space has property X iff it can be embedded in itself in this way? My intuition says no, but I can't think of a counterexample offhand, partly because I don't know any interesting non-compact metric spaces.

2. Is property X equivalent to not being compact? If every non-compact (completely metrizable) space has a metrizable compactification then this is easy, but I don't know whether that is the case, although I suspect it might be. Which leads onto the following weaker question:

3. Given a space which does not have property X, can it be embedded in a space which does? Here my intuition says yes, this should be the case, but I can't think of a more general approach for constructing a suitable embedding space than compactification which, as I said, I can't prove will give a metrizable space. I don't think I know enough topology (I've only taken a couple of basic undergraduate courses).

Edit: 3. above should read "given a space which has property X, can it be embedded in a space which does not?

Answer 1

The answer to 2 is yes. Let $(X,d)$ be complete but not compact, then for some $r>0$ it has a countable r-separated subset $S=\{s_i\}:i=1,2,\dots$. Let $d'$ be the maximal metric on $X$ satisfying $d'\le d$ and $d'(s_i,s_j)\le r/\min(i,j)$ for all $i,j$. Then $(X,d')$ is homeomorphic to $(X,d)$ - in fact, the identity map is a local isometry - but $d'$ is not complete. So we have embedded the space into the completion of $(X,d')$ as a non-closed subset.

The metric $d'$ can be constructed explicitly: $d'(x,y)$ is the minimum of $d(x,y)$ and the infimum of sums $d(x,s_i)+d(x,s_j)+r/\min(i,j)$ over all pairs of $i,j$. Verifying the triangle inequality is straightforward.

As for 3, the answer is no, because you cannot embed any complete space into a compact space. For example, a non-separable Banach space cannot be so embedded, as Qiaochu Yuan explained in comments.

Update. It seems that I misunderstood Q3. As stated, it asks whether every compact space can be embedded into a complete non-compact one. The answer is of course yes, as Ady noticed.

Answer 2

Any metric space can be isometrically embedded into some Banach space E. And E has the property X, since it can be topologically embedded into itself as its open unit ball, which is clearly not closed in E. Therefore, the answer to 3. is YES, and the intuition of Robin Saunders is very good.

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This is just to clarify some things. For this, let $(M,d)$ be a complete metric space.

It is pretty clear that a compact metric space cannot have "the property X". Conversely, asumming that $M$ is non-compact and that $d$ is bounded, by taking into consideration the equivalent metric $d^{*}\left(x,y\right)$= $\sup_{t\in M}$ $\mid f(x)\cdot d(x,t)$ $-f\left(y\right)\cdot d\left(y,t\right)\mid$, where $f:M$ $\rightarrow$ $(0,\infty)$ is bounded and continuous and inf$_{M}$ $f$ = 0, one easily obtain that $(M,d^*)$ is not complete.

Consequently, $M$ has "the property X" iff it is non-compact.

Now, every separable metric space can be embedded into the Hilbert cube, which has not "the property X" .

[Note that on Wikipedia you'll find "every separable metric space can be isometrically embedded into the Hilbert cube", which is obviously not true.]

OTOH, any subspace of a separable metric space is separable, too. And any compact metric space is, clearly, separable.

Therefore, $M$ is embeddable into a space having not "the property X" iff $M$ is separable.

Answer 3

The answer to 1. is definitely no: a regular tree of valency at least $3$ is a counter-example. An embedding into itself must have closed image due to the combinatorial constraints. However, it can easily be embedded into a plane with non-closed image.