Is Every Point in a Topological Space Closed?

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Homework Help Overview

The discussion revolves around the question of whether every point in a topological space is closed, with specific references to metric spaces and T1 spaces.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants suggest constructing counterexamples, particularly considering single and two-point spaces. There are mentions of T1 spaces, Hausdorff spaces, and discrete topology as avenues for exploration.

Discussion Status

Some participants have provided guidance on exploring specific types of topological spaces, while others have raised questions about the validity of the original statement. Multiple interpretations of the problem are being explored, particularly regarding the nature of closed sets in different topological contexts.

Contextual Notes

There is an ongoing discussion about the implications of different topological properties, such as T1 and indiscreet topology, and how they relate to the original question. The completeness of definitions and examples is still under examination.

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Homework Statement


Is it true that every point in a topological space is closed? In a metric space?



Homework Equations





The Attempt at a Solution

 
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Have you tried constructing a counterexample?

A single point space clearly won't suffice; what about a two-point space?
 
Try looking up T1 spaces.
Then look at Hausdorff spaces and metric spaces and try to prove whether they are in general T1.
 
you can also think about discrete topology, if you have learned before.
 
Hurkyl said:
Have you tried constructing a counterexample?

A single point space clearly won't suffice; what about a two-point space?

So, I found it is only true in a T1 space.

But in a single point space it is also true since the complement of every point is the null set which is open.
 
Actually, it might be better to think about the indiscreet topology rather than the discreet topology.
 

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