Can anyone give an example of such continuous injection?

Question

An example of a continuous injection from a rectangle in R^2 to R (a point, of course, is not considered a rectangle, or it'd be trivial).

And continuous bijections, are they posible?

Thank you

Answer 1

No. No one can. Such injection doesn't exist. Let's show this in a general context.

Proposition:

Let T be a connected topological space composed of at least 2 points. Suppose that "punctures" on T preserve connectedness, that is, for every t in T, the set T(t) = T - {t} is connected. Then, there's no injection (hence, no bijection) from T to R.

Proof:

Suppose f is a continuous injection from T to R. Then, f(T) is connected (continuous mappings preseve connectedness) and contains at least 2 elements (because f is injective). The connected subsets of R are the intervals, including the so called degenerated intervals of the form [x,x] = {x}. Since this last case is discarded, it follows f(T) contains a non-empty open interval.

Let a be an interior point of f(T) and t the only element of T such that f(t) = a. Since, by assumption, T(t) = T - {t) is connected, so is f(T(t)), because f, being continuous on T, has a continuous restriction to T(t) (which, therefore, preserves connectedness). In addition, the injective character of f implies that f(T(t)) = f(T) - {a). But since f(T) is an interval and a is one of its interior points, it follows(-oo, a) and (a, oo) form a disconnection of F(T(t)), contradicting the previous conclusion that f(T(t)) was connected.

Hence, there's no continuous injections - let alone bijections - from T to R.

Since rectangles of R^2 (excluding singletons) meet all the requirements of the proposition, there's no continuous injection from them to R. The same is true for disks and regions, in general (connected open sets), as well as for their closures. Not only in R^2, but in R^n, n>=2

But if you drop continuity, then it's possible to find injections and sometimes even bijections from regions in R^2 to R.

Answer 2

Steiner's answer is of course sufficient, but it is in fact interesting that there exists bijections from R^n to R for all values of n!