(a,b) not homeomorphic to [a,b]

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In summary, to prove that (0,1) is not homeomorphic to [0,1], one can remove a point from each space and check for connectedness. The topology used is assumed to be the standard topology unless otherwise specified. Another method is to show that one space is compact while the other is not.
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JG89
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Homework Statement


Show that (0,1) is no homeomorphic to [0,1]


Homework Equations





The Attempt at a Solution



I understand the usual procedure for proving this - remove a point from each of the spaces and check for connectedness. My only question is, the topology on each space isn't specified. Do I assume it's the subspace topology or what?

In general, when given questions like this, with no topology given, do I just infer from the context of the question which topology I should use?
 
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  • #2
Nevermind guys. Looks like I need to pay more attention. My book says to assume the standard topology unless otherwise noted.
 
  • #3
I understand the usual procedure for proving this - remove a point from each of the spaces and check for connectedness.

This is a pretty awesome way of proving they're not homeomorphic to be honest. I think most people would just show that one is compact and the other is not
 

1. What does it mean for two sets to be homeomorphic?

Homeomorphism is a concept in topology that describes a continuous mapping between two topological spaces. Essentially, it means that the two spaces are structurally equivalent, and any property that can be defined on one space can also be defined on the other.

2. Why can't (a,b) and [a,b] be homeomorphic?

Intuitively, (a,b) and [a,b] are not homeomorphic because they have different "shapes". (a,b) is an open interval, meaning that it does not include its endpoints, while [a,b] is a closed interval, meaning that it includes both endpoints. These differences in the boundary points and openness cannot be preserved in a continuous mapping, making them non-homeomorphic.

3. Can (a,b) and [a,b] be homeomorphic if we consider them as subsets of the real numbers?

No, even if we consider (a,b) and [a,b] as subsets of the real numbers, they cannot be homeomorphic. This is because the real numbers have a property called connectedness, which means that any two points can be connected by a continuous curve. However, in (a,b) and [a,b], the endpoints a and b act as barriers, preventing the two sets from being connected in this way.

4. Are there any other examples of sets that are not homeomorphic?

Yes, there are many other examples of sets that are not homeomorphic. Some common examples include a circle and a square, a sphere and a cube, and a line segment and a point. Generally, any two sets with different dimensions or topological properties will not be homeomorphic.

5. Is homeomorphism the only way to compare the structures of two sets?

No, homeomorphism is just one way to compare the structures of two sets. There are other concepts in topology, such as homotopy and homology, that can also be used to compare sets and spaces. These concepts look at different aspects of the sets, such as their continuous deformations or their algebraic properties, to determine if they are equivalent in some way.

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