What is an "induced map"? Is it a Quotient Map?

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Discussion Overview

The discussion revolves around the concept of an "induced map" in the context of topology, specifically relating to functions between spaces and the formation of quotient spaces. Participants explore the properties of the induced map and its relation to quotient maps, as well as the implications of certain functions and equivalence relations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on the definition of an "induced map" and questions whether it is a quotient map, noting that their function i is not necessarily an inclusion.
  • Another participant suggests that the induced map f is simply the identity map on B and proposes that the disjoint union of B and X is taken modulo the equivalence relation defined by g(a) = i(a).
  • A different participant agrees with the previous assertion, indicating that the equivalence relation allows for the identification of points in B and X, leading to a glued space that likely has the quotient topology.
  • Questions arise regarding the concept of cofibrations, with one participant defining it in terms of the homotopy extension property and its relation to the map i.
  • There is a reiteration of the understanding of the map's function, suggesting that the construction aligns with the previous interpretations.

Areas of Agreement / Disagreement

Participants express some agreement on the nature of the induced map and its relationship to the quotient topology, but there remains uncertainty regarding the precise definitions and implications of the terms involved. The discussion does not reach a consensus on the characterization of the induced map or the nature of the quotient space.

Contextual Notes

There are unresolved aspects regarding the definitions of induced maps and quotient spaces, as well as the implications of the functions involved. The discussion also touches on the concept of cofibrations, which may introduce additional complexity to the topic.

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I have what I hope to be just a simple notation/definition question I can't seem to find an answer to.

I'm not going to post my homework question, just a piece of it so I can figure out what the question is actually asking. I have a function i:A --> X I also have a continuous function g: A --> B. Then I am asked to prove a property about the "induced map" f: B --> B Ug XI am just having trouble understanding exactly what this "induced map" is. There are no defs for it in my book and online I only see induced map as induced homeomorphisms. So my question: is this a quotient map? Is B Ug X just a quotient space? My map i is not necessarily an inclusion, so A and X could be two separate spaces, so I"m assuming this is a quotient space because an a in A could map to X under i but could also map to B under g. I guess I'm just confused about this function. Does i even factor into this map?Also when I think of the composition f(i(A)) what on Earth this would be like.
 
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dumbQuestion said:
I have a function i:A --> X I also have a continuous function g: A --> B. Then I am asked to prove a property about the "induced map" f: B --> B Ug X

Take the disjoint union of B and X modulo the equivalence relation g(a)= i(a).

The f is just the identity map on B.

This seems misstated? are you sure this is right?
 
the only thing I could see is that i allows you to form the equivalence relation. When you identify i(a) and g(a) you glue B to X. This glued together space probably has the quotient topology.

What is a cofibration?
 
I think you're right about what the map is doing, that's what I was thinking myself.

cofibration is another mess... its just a map that has the homotopy extension property for every space Y. So, i : A --> X is a cofibration if, for every homotopy H_t : A --> Y and every map M: X --> Y that agrees with H_0 on A, then you can extend the homotopy H_t to one that goes from X --> Y and agrees on A.
 
dumbQuestion said:
I think you're right about what the map is doing, that's what I was thinking myself.

cofibration is another mess... its just a map that has the homotopy extension property for every space Y. So, i : A --> X is a cofibration if, for every homotopy H_t : A --> Y and every map M: X --> Y that agrees with H_0 on A, then you can extend the homotopy H_t to one that goes from X --> Y and agrees on A.

then i think the construction is as we think.
 

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