Question

Homeomorphism in topological spaces give an example

Answer 1

Answer:

Show that the topological spaces (0,1) and (0,∞) (with their topologies being the unions of open balls resulting from the usual Euclidean metric on these subsets of R) are homeomorphic.

To show that these two topological spaces are homeomorphic we must find a continuous bijection f:X→Y such that f−1 is also continuous.

Consider the following function f:(0,1)→(1,∞) given by:

(1)

f(x)=1x

We first show that f is bijection. Let x,y∈(0,1) and suppose that f(x)=f(y). Then:

(2)

1x=1y

Cross multiplying gives us that then x=y, so f is injective.

Now let b∈(1,∞). Since b>1 we have that 0<1b<1, and so let a=1b. Then:

(3)

f(a)=1a=11b=b

So for all b∈(1,∞) there exists an a∈(0,1) such that f(a)=b, so f is surjective.

It's not hard to see that f is a continuous map. Furthermore, f−1:(1,∞)→(0,1) is also given by f−1(x)=1x (which is continuous), and so f is a homeomorphism between (0,1) and (1,∞), so these spaces are homeomorphic.

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