Homeomorphism in Topologies

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rawanok

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Hi. :)

I need your help about a question. Thanks in advance.

"Give standard topological space on R2. Give another topological space example that is not homeomorphic to this topological space on R2 and state its causes using theorems."
 
rawanok said:
Hi. :)

I need your help about a question. Thanks in advance.

"Give standard topological space on R2. Give another topological space example that is not homeomorphic to this topological space on R2 and state its causes using theorems."
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Please make sure your question is Exactly as it was given to you. You seem to be missing some words! In addition:

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.
 
rawanok said:
"Give standard topological space on R2. Give another topological space example that is not homeomorphic to this topological space on R2 and state its causes using theorems."
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Can you at least give us the description of the "usual" topology on \(\Re^2\)?
What is the uncountable discrete topology on \(\Re^2\)?
Are those two entirely different?
You need to go to your college's mathematics library and use COUNTEREXAMPLES IN TOPOLOGY by Lynn Steen & Art Seebach.
 
rawanok said:
For example, the unit can be a circle. View attachment 19455
How can I create non-homeomorphic topologies to this topology?
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I repeat my request: Can you at least give us the description of the "usual" topology on \(\Re^2\)?
If you cannot the we must conclude that you know very very little about topology and you are trolling.
 
pka said:
I repeat my request: Can you at least give us the description of the "usual" topology on \(\Re^2\)?
If you cannot the we must conclude that you know very very little about topology and you are trolling.
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Consider the Cartesian plane R2, then the collection of subsets of R2 which can be expressed as a union of open discs or open rectangles with edges parallel to the coordinate axis from a topology, and is called a usual topology on R2.
 
rawanok said:
Consider the Cartesian plane R2, then the collection of subsets of R2 which can be expressed as a union of open discs or open rectangles with edges parallel to the coordinate axis from a topology, and is called a usual topology on R2.
Click to expand...
@rawanok, I hope that you will forgive me for doubting that you are in a real topology class. Can you give us the name of your textbook? I probably have it, if not our library will.
If \(\bf{T}=\{\Re^2,\emptyset\}\) , then \(\bf{T}\) is a topology on \(\Re^2\) what is its name?
If \(\bf{T}=\mathscr{P}(\Re^2)\), the powerset of \(\Re^2\). Then \(\bf{T}\) is a topology on \(\Re^2\) what is its name?
 
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