Separated topology and existence of a metric

In summary, there is a well-known theorem that states when a space is metrizable, but there may be restrictions on the cardinality or second countability for certain spaces to be metrizable. Additionally, in order for a space to be metrizable, it must also be Hausdorff and have certain properties such as being second countable.
  • #1
seratend
318
1
Can we proove that for any separated topological space, there exists a metric?

Seratend.
 
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  • #2
No, I don't think so. THere is a well known theorem that states when a space is metrizable. Try googling for it.
 
  • #3
Appears you must have some kind of restriction on the cardinality of some things.

(exactly what do you mean by separated?)
 
  • #4
matt grime said:
Appears you must have some kind of restriction on the cardinality of some things.

(exactly what do you mean by separated?)

sorry: direct french translation.
for any two different points (x,y) of this set, I have at least two disjoint open sets (A,B), such that x element of A and y element of B.

And yes, I think this is theorem I have forgotten about metrizable spaces, I am searching it now again.

Seratend
 
  • #5
oh, Hausdorff.

you need second coutable (if it is compact) so something like a product of [0,1] indexed by some very large cardinal won't be metrizable.

see also Uhyrson's lemma
 

1. What is separated topology?

Separated topology is a mathematical concept that defines a way of organizing and studying the properties of a set by considering its subsets and their relationships.

2. How does separated topology differ from other types of topology?

Separated topology is distinguished from other types of topology by the requirement that the subspaces of a set must be disjoint, meaning they have no points in common. This allows for a more precise analysis of the structure and properties of the set.

3. How is the existence of a metric related to separated topology?

The existence of a metric, or a way to measure distances between points, is essential for separated topology to be defined. A metric provides a way to quantify the separation between points and determine if a set satisfies the requirements for separated topology.

4. Can separated topology be applied to real-world problems?

Yes, separated topology has numerous applications in fields such as physics, engineering, and computer science. It can be used to analyze and solve problems involving complex systems, networks, and data structures.

5. Are there any limitations to using separated topology?

Like any mathematical concept, separated topology has its limitations and may not be applicable to every problem or situation. It is important to carefully consider the specific properties and requirements of a problem before using separated topology as a tool for analysis.

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