Which of the following topologies are metrizable?

[math25]

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

 

[micromass]

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??

 

[math25]

they are always Hausdorff ?

Am I right for e) a) and b) ?

 

[math25]

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?

 

[sunjin09]

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)

 

[math25]

thanks, what do you think about b) please see the post above yours

 

[sunjin09]

thanks, what do you think about b) please see the post above yours

I think your reasoning is right, since R is uncountable.

 

[micromass]

Is e Hausdorff??

 

[math25]

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?