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

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Homework Help Overview

The problem involves finding two compact spaces, A and B, that are nonhomeomorphic yet satisfy the condition that the product spaces AX[0,1] and BX[0,1] are homeomorphic. The subject area pertains to topology, specifically the study of homeomorphisms and compact spaces.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss potential candidates for spaces A and B, such as the unit interval [0,1] and the circle S^1. There is uncertainty about the homeomorphic relationship between the unit square and the cylinder. One participant suggests a hint regarding the nature of the homeomorphism and its implications for the values of t and s. Another participant considers the concept of homotopy in relation to the hint provided.

Discussion Status

The discussion is active with participants exploring various ideas and questioning the relationships between the proposed spaces. A hint has been offered that seems to clarify the nature of the homeomorphism, prompting further exploration of related concepts like homotopy. There is no explicit consensus yet, but the dialogue is productive.

Contextual Notes

Participants are grappling with definitions and properties of homeomorphisms, and there is a mention of cardinality as a possible consideration. The problem's constraints and the requirement for compactness are acknowledged but not resolved.

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Homework Statement



Find two spaces A,B compact where A and B are nonhomeomorphic but AX[0,1][tex]\cong[/tex]BX[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.
 
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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
 
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?
 
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?
 

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