Can Bounded, Constant Functions from [a,b] be Homeomorphisms to the Real Line?

[qinglong.1397]

Homework Statement

Let A be a collection of bounded, constant functions from the interval [a, b] to the real line.

Then, is there a homeomorphism from A to the real line?



Thanks!

 

[fzero]

You should show some work. Start with the definition of a homeomorphism.

 

[qinglong.1397]

You should show some work. Start with the definition of a homeomorphism.

Oh, yeah, I should.

First, assign set A a metric which is defined as $$\rho$$(f,g)=sup|f(x)-g(x)| for x in [a, b]. Then A is a metric space.

Second, the real line has the usual topology.

So we may find a homeomorphism between these two topological spaces? I think the answer is Yes.

Since every function f in A should be bounded, that is, the value of f should be finite, the first conclusion I can get is that the set B of the values of the functions in A seems bounded. But if B is bounded, we can find a value g larger than f in A and g(x) would be a constant function on [a, b] with definite value, so g is contained in A. Then A is enlarged and B enlarged, too. We can repeat similar process infinitely and in the end B=R. So I think there is a homeomorphism.

But there is a problem. Since B=R, there is a function equal +infinity and the other one -infinity. These two functions are not bounded. So then, B does not equal R. If so, we can always find a upper bound of B and a lower bound, then similar argument will appear again which leads to that B=R.

What is the problem?