Lower Limit Topology: Explaining Lemma 13.4 of Munkres' Topology

[princy]

how does a lower limit topology strictly finer than a standard topology? please explain lemma 13.4 of munkres' topolgy..

 

[radou]

How do we define the relation "to be finer"? What does it mean?

 

[radou]

The answer to your question, i.e. the proof of Lemma 13.4. is a direct application of Lemma 13.3.

 

[Tinyboss]

Can you write a basis element of the standard topology as a union of basis elements in the lower limit topology? Hint: yes.

 

[princy]

Can you write a basis element of the standard topology as a union of basis elements in the lower limit topology? Hint: yes.

thanks for ur hint.. i got the idea.. am now not able to prove it for k-topology... that is.. how is it possible that the basis element of standard topology is the basis element of k-toplogy??..
it is stated that.."given a basis element of (a,b) of T and a point x of (a,b),this same interval is a basis element for T'' "...
i'm not getting it clear.. can u help me in this??

 

[princy]

thanks for ur hint.. i got the idea.. am now not able to prove it for k-topology... that is.. how is it possible that the basis element of standard topology is the basis element of k-toplogy??..
it is stated that.."given a basis element of (a,b) of T and a point x of (a,b),this same interval is a basis element for T'' "...
i'm not getting it clear.. can u help me in this??

thank u for ur help.. i got it clear that it was proved on the basis of lemma 13.3...

 

[princy]

k-topology

if k-topology is defined as {(a,b)-k : k={1/n,n=1,2,3,...},(a,b) is in R}, can i write its elements as (a,b)-(0,1]?. what i have understood is this k-topology has the basis element which excludes the points {1,1/2,1/3...} which is the interval (0,1].. so from this how can i prove that this basis is the same as that of the basis element of a standard topology?