Is $\Bbb R$ homeomorphic to a cartesian product?

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  • Thread starter Euge
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    2017
In summary, a homeomorphism is a type of continuous function between two topological spaces with an inverse function that is also continuous. Two spaces that are homeomorphic can be considered equivalent in terms of their shape or structure. There is a difference between homeomorphism and isomorphism, with the former specifically referring to continuous functions between topological spaces. To determine if two spaces are homeomorphic, one can look for a function that preserves topological properties. The significance of $\Bbb R$ being homeomorphic to a cartesian product is that the real numbers can be represented in terms of two other topological spaces, which has various applications in mathematics.
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Euge
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Here is this week's POTW:

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Prove that if $\Bbb R$ is homeomorphic to a cartesian product $A\times B$, then either $A$ or $B$ is a singleton.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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No one answered this week's problem. You can read my solution below.
Let $(a,b)$ be a point of $A \times B$ corresponding $0$ under a homeomorphism $\Bbb R \to A \times B$. Then there is a homeomorphism $\Bbb R\setminus \{0\} \to (A\times B) \setminus \{(a,b)\}$. Therefore, $(A \times B)\setminus\{(a,b)\}$ is disconnected. Since $A \times B$ is connected, then $A$ and $B$ are connected. So if neither $A$ nor $B$ is a singleton, then $(A\times B)\setminus\{(a,b)\}$ is connected. ($\rightarrow\leftarrow$)
 

FAQ: Is $\Bbb R$ homeomorphic to a cartesian product?

What is a homeomorphism?

A homeomorphism is a type of continuous function that exists between two topological spaces and has an inverse function that is also continuous.

Can two spaces that are homeomorphic be considered the same?

Yes, if two spaces are homeomorphic, it means that they have the same topological structure and can be considered equivalent in terms of their shape or structure.

Is there a difference between homeomorphism and isomorphism?

Yes, while both terms describe a type of mapping between two mathematical structures, homeomorphism specifically refers to continuous functions between topological spaces, whereas isomorphism can refer to various types of structures such as groups, rings, or vector spaces.

How can I tell if two spaces are homeomorphic?

One way to determine if two spaces are homeomorphic is to look for a function that can map points from one space to the other while preserving their topological properties, such as continuity and connectivity.

What is the significance of $\Bbb R$ being homeomorphic to a cartesian product?

This means that the real numbers can be represented as a product of two other topological spaces. This has various applications in mathematics, such as in the study of manifolds and functional analysis.

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