# Lower limit topology

## Main Question or Discussion Point

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

## Answers and Replies

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

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

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

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??

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...

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?