Homeomorphism in Topologies

[rawanok]

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."

 

[Deleted member 4993]

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."

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.

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[pka]

"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."

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]

@pka @Subhotosh Khan
For example, the unit can be a circle.


How can I create non-homeomorphic topologies to this topology?

 

[pka]

For example, the unit can be a circle. View attachment 19455
How can I create non-homeomorphic topologies to this topology?

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.

 

[rawanok]

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.

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.

 

[HallsofIvy]

@pka @Subhotosh Khan
For example, the unit can be a circle.
View attachment 19455

How can I create non-homeomorphic topologies to this topology?

No, this is a single curve, it is not a topology!

 

[pka]

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, 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?