Lower Limit Topology: Explaining Lemma 13.4 of Munkres' Topology

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Discussion Overview

The discussion focuses on the lower limit topology and its relationship to the standard topology, specifically in the context of Lemma 13.4 from Munkres' Topology. Participants explore definitions, proofs, and examples related to these topological concepts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks how the lower limit topology is strictly finer than the standard topology and requests an explanation of Lemma 13.4.
  • Another participant inquires about the definition of "to be finer" in the context of topologies.
  • A participant suggests that the proof of Lemma 13.4 is a direct application of Lemma 13.3.
  • Several participants discuss whether a basis element of the standard topology can be expressed as a union of basis elements in the lower limit topology, with hints provided that it is possible.
  • One participant expresses confusion regarding proving the relationship between basis elements of the standard topology and k-topology, specifically referencing intervals and points excluded in k-topology.
  • Another participant reiterates their confusion about the basis elements of k-topology and seeks clarification on how they relate to the standard topology.
  • A participant mentions that they have understood the proof is based on Lemma 13.3 but still seeks further clarification on the relationship between k-topology and standard topology basis elements.
  • A participant defines k-topology and discusses its basis elements, questioning how to prove that these elements correspond to those of the standard topology.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the relationships between different topologies and their basis elements. There is no consensus on the clarity of the proofs or definitions discussed, indicating ongoing confusion and exploration of the topic.

Contextual Notes

Participants reference Lemma 13.3 and its application to Lemma 13.4, but there are unresolved questions about the definitions and relationships between the standard topology, lower limit topology, and k-topology. Specific assumptions about the nature of basis elements and their intersections are not fully clarified.

princy
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how does a lower limit topology strictly finer than a standard topology? please explain lemma 13.4 of munkres' topolgy..
 
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How do we define the relation "to be finer"? What does it mean?
 
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.
 
Tinyboss said:
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 said:
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?
 

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