Topology Problem: Find 2 Nonhomeomorphic Compact Spaces AX[0,1]≅BX[0,1]

[tt2348]

Homework Statement

Find two spaces A,B compact where A and B are nonhomeomorphic but AX[0,1]\congBX[0,1]

Homework Equations

Definitions of homeomorphism, cardinality possiby, I have no idea where to start.

The Attempt at a Solution

My idea Is [0,1] and S^1, but I am not sure if the unit square is homeo to the cylinder.

 

[Office_Shredder]

It's not. The cylinder has a loop in it that can't be shrunk to a point (one going around it) and the unit square doesn't have anything like that.

A hint: If f(x,t) is the homeomorphism from Ax[0,1] to Bx[0,1], it must be that if f(x,t)=(y,s) that t is not equal to s in general, otherwise restricting yourself to a single value of t would give a homeomorphism from A to B

 

[tt2348]

That's actually a really useful idea. For some reason I look at your hint and think homotopy. Is that a step in the right direction?

 

[tt2348]

How about A= space between two concentric circles, with a smaller circle glued to the inside of the smaller circle, and a circle of the same size glued to the outside, and B= the same concentric circle space, but two circles are on the outside?