1

At first I want to say that I'm aware of that there is alot of threads regarding this topic. But I would like to Believe that my question differs from them.

So I'm given the claim that there is a complete metric space $(X, d_X)$ and a non complete metric space $(Y,d_Y)$. Naturally I to verify the claim I want to come up with an example. My question then is,

What would be an natural thought process to come up with such an example?

My thoughts Went something like this.

  • The function $f$ is a bijection, okey so I need to find two sets with the same cardinality.

  • $f$ is continuous, perhaps and can use some elementary function, and nothing to "weird"

  • the inverse function $f^-1$ is continuous, again perhaps I can use something elementary.

I know that $\mathbb{R}$ with $d(x,y) = |x-y|$ is complete, and I know that $(0,1)$ and $\mathbb{R}$ has same cardinality. How can I proceed to find a function $f$? Is it just trial and error, or is the a smart way?

Olba12
  • 2,579

1 Answers1

1

You've correctly identified the two "obvious" spaces you want to show are homeomorphic.

To find a homeomorphism: note that a strictly increasing continuous function on an interval is bijective onto its image and moreover has continuous inverse. Therefore it's enough to find a strictly increasing surjective continuous function $f: (0,1) \to \mathbb{R}$. It's intuitively really obvious that such functions exist; noting that something based on $\tan^{-1}$ works is just a matter of spotting the neatest solution. Other homeomorphisms are available, though they might not be as neat to state.

For example, my first thought would be to do some kind of piecewise-linear thing which is rotationally symmetric about the point $(\frac{1}{2}, 0)$, where the gradient changes every $x=\frac{1}{2^n}$ and $1-\frac{1}{2^n}$, sloping steeper and steeper (gradient $e^{e^n}$ or something, I can't be bothered to make it sharp) as $n$ increases.

Patrick Stevens
  • 36,135