Two metrics induce the same topology, but one is complete and the other isn't

I'm looking for an example of two metrics that induce the same topology, but so that one metric is complete and the other is not (Since it is known that completeness isn't a topological invariant).

Thanks in advance for any hints or ideas.

Solution 1:

The metric space $\{\frac {1}{n} \mid n\in \mathbb{N} \}$ with the usual metric is incomplete (since we don't have zero), and it has the discrete topology.

the same space with the metric $d(x,y)=1 \iff x\neq y$ also has a discrete topology but is complete, since any cauchy sequence will eventually be constant.

Solution 2:

The interval $M:=]-1,1[$ with the usual line element $ds:=|dx|$ is an incomplete metric space: The sequence $x_n:=1-{1\over n} \ (n\to\infty)$ converges in $\mathbb R$, so it is a Cauchy sequence, but it diverges in $M$. On the other hand, the "hyperbolic metric" defined by $ds:=|dx|/(1-x^2)$ induces the same topology on $M$ but is complete. The latter statement needs of course a proof. Suffice it here to say that now the endpoints $\pm1$ are "infinitely far away".

Solution 3:

Look for a homeomorphism $f: \mathbb R \to \mathbb R_+$ (you should know a good one). Then take the usual metric on $\mathbb R$ and use it to define a metric $d$ on $\mathbb R_+$ by $d(x,y) = |f(x)-f(y)|$ (*). Now what can you say about the relationship between $d$ and the usual metric on $\mathbb R_+$?

EDIT: Made the question here correct and more illustrative.

So the identity $\iota: (\mathbb R_+,d) \to (\mathbb R_+,|\cdot|)$ is a homeomorphism. Now just ask yourself which property, that continuous functions between metric spaces can have, $\iota$ doesn't have? Can you formulate a theorem about what property a homeomorphism must have to preserve completeness?

(*): Pullback and pushforward might be of interest.

Solution 4:

As many others have pointed out, the way to do this is to think about pairs of metric spaces that are homeomorphic but not isometric. In many cases, the metric will be complete on one space but not the other. For example:

• The set $\{1/n \mid n \in \mathbb{N}\}$ is incomplete, but the natural numbers $\mathbb{N}$ are complete.

• The ray $(0,\infty)$ and the open interval $(0,1)$ are incomplete, but the real line is complete.

• An open disc in the plane is incomplete, but the entire plane is complete.

• The punctured plane $\mathbb{R}-\{0\}$ is incomplete, but the infinite cylinder $\mathbb{R} \times S^1$ (where $S^1$ is the circle) is complete.

• The real line minus the $x$-axis is incomplete, but a disjoint pair of planes is complete.

• The twice-punctured plane $\mathbb{R}^2 - \{(-1,0),(1,0)\}$ is incomplete, but the graph of $$ z = \frac{1}{(x^2-1)^2 + y^2} $$ in $\mathbb{R}^3$ is complete. (This graph has asymptotic "cusps" at $(-1,0)$ and $(1,0)$.)

In each case, the homeomorphism between the two spaces can be used to define a nonstandard metric on the latter space that makes it incomplete.