# Thread: Topology Question...

1. ## Topology Question...

Hi guys,

I really need help with the following question:

Let A be the infinite countable union of circles with radius 1/n & centres (1/n,0) in Euclidean 2-space. ie - subspace

Let B be the infinite countable union of circles with radius n & centres (n,0) in Euclidean 2-space. e - subspace

Let C be the wedge of countably many circles.

1) Which of these 3 spaces are CW complexes?

2) Attempt to construct contiinuous bijectiive maps between these spaces (if possible) & then show if these are homeomorphisms.

3) Are the any other more "immediate" way of showing any of the spaces are homeomorphic?

I'm completely stuck! :-( Any assistance would be ENORMOUSLY helpful!

Thanks in advance. :-) x

2. PS - the instructor hinted that A is not homeomorphic to B or C - but I have no idea how to show this! :-s :-(

Thanks x

3. Originally Posted by TTB
Hi guys,

I really need help with the following question:

Let A be the infinite countable union of circles with radius 1/n & centres (1/n,0) in Euclidean 2-space. ie - subspace

Let B be the infinite countable union of circles with radius n & centres (n,0) in Euclidean 2-space. e - subspace

Let C be the wedge of countably many circles.

1) Which of these 3 spaces are CW complexes?

2) Attempt to construct contiinuous bijectiive maps between these spaces (if possible) & then show if these are homeomorphisms.

3) Are the any other more "immediate" way of showing any of the spaces are homeomorphic?

I'm completely stuck! :-( Any assistance would be ENORMOUSLY helpful!

Thanks in advance. :-) x
Every CW complex is locally contractible, but your space A is not locally contractible at 0. A is also compact. Remember that "compactness" is a topological invariant. If one space is compact and the other space is not compact, we can't find a homeomorphism between them.