Which of the following topologies are metrizable?

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In summary, topology a) is not metrizable as it is not Hausdorff. Topology b) is also not metrizable because it is not Hausdorff, as the cocountable topology is not Hausdorff. On the other hand, topologies c) and d) are metrizable because they are Hausdorff. Lastly, topology e) is metrizable because it is Hausdorff. Some other properties of metrizable spaces include being regular, normal, and second countable.
  • #1
math25
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Can someone please help me to determine which of the following topologies are metrizable?

a)Let X be any inf nite set and let T = {U subset of X : x\u is finite }

b) Let X = R and let T = {U subset of R : R \ U is FI nite of countable }

c) For each k in N, let Nk = {1; 2,...,K} . Let T = {empty set} U{N}U {Nk: k is in N}

d) For each k in N, let U = { k; k + 1; k + 2;...} then T = { empty set} U {Uk : k is in N}

e) Let T = {empty set} U { R} U { (a, infinity) : a is in R}


I think that {e} is not metrizable (weak topology and not countable)
Also, I think that a) and b) are metrizable , however not sure about c) and d)

thanks
 
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  • #2
Can you list some properties of metrizable spaces??

For example, I know that in a metrizable space

1) Every singleton is closed
2) It is first countable
3) ...

Can you list some more properties??
 
  • #3
they are always Hausdorff ?

Am I right for e) a) and b) ?
 
  • #4
Actually, I think a) is not metrizable because it is not Hausedorff ?

Also, for b) the cocountable topology is not Hausedorff therefore b) is not metrizable?
 
Last edited:
  • #5
math25 said:
Actually, I think a) is not metrizable because it is not Hausedorff ?

You are right. That finite complement topology is the first non-metrizable space I ever know. Similar for (b)
 
Last edited:
  • #6
thanks, what do you think about b) please see the post above yours
 
  • #7
math25 said:
thanks, what do you think about b) please see the post above yours

I think your reasoning is right, since R is uncountable.
 
  • #9
Well, before I thought it wasn't, but I am not sure anymore...it seems like it is...

c) and d) are Hausdorff, therefore metrizable, right?
 

Related to Which of the following topologies are metrizable?

1. What does "metrizable" mean in topology?

Metrizable refers to a topological space that can be described by a metric, which is a mathematical concept that measures distance between points in a space. In other words, a metrizable topology is one that can be understood in terms of distances between its elements.

2. What are the criteria for a topology to be metrizable?

In order for a topology to be metrizable, it must satisfy certain conditions, such as being Hausdorff, second countable, and regular. These conditions ensure that the topology can be described by a metric and that it follows certain properties related to distance and continuity.

3. Which topologies are considered metrizable?

Some common examples of metrizable topologies include the Euclidean, metric, and discrete topologies. However, any topology that satisfies the necessary criteria can be considered metrizable, and there are many different types of metrizable topologies.

4. What is the importance of metrizability in topology?

Metrizability is important in topology because it allows for a more concrete understanding of the space being studied. By describing a topology in terms of a metric, it becomes easier to make calculations and prove theorems about its properties. Additionally, metrizable topologies often have simpler and more intuitive structures, making them easier to work with.

5. Are all topologies metrizable?

No, not all topologies are metrizable. In fact, the majority of topologies are not metrizable. This is because there are many different ways to define a topology, and not all of them can be described by a metric. However, many commonly studied and useful topologies are metrizable.

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