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

In summary, the conversation discusses the concept of an induced map in the context of continuous functions and quotient spaces. The "induced map" is defined as the identity map on the disjoint union of two spaces, B and X, modulo the equivalence relation g(a)= i(a). It is also mentioned that this construction may have the quotient topology. The conversation also briefly touches on the definition of a cofibration, which is a map that has the homotopy extension property for every space Y. The speaker agrees that the construction of the induced map is as they had initially thought.
  • #1
dumbQuestion
125
0
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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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 
  • #5
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.
 

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

What is an "induced map"?

An induced map, also known as a pushforward or a lift, is a function between two mathematical spaces that is defined by mapping each point in the first space to a corresponding point in the second space using a given transformation or mapping rule.

What is the difference between an induced map and a quotient map?

An induced map is a function between two mathematical spaces, while a quotient map is a function between two topological spaces. An induced map is defined by a mapping rule, while a quotient map is defined by a set of equivalence relations. Additionally, an induced map preserves the structure of the first space, while a quotient map collapses the structure of the first space into the second space.

What is the purpose of an induced map?

The purpose of an induced map is to extend a given function between two spaces to a larger space by mapping each point in the original space to a corresponding point in the larger space. This allows for the study of a larger, more complex space by using the properties and relationships of the original space.

What are some examples of induced maps?

One example of an induced map is the projection map in linear algebra, which maps a vector space onto its quotient space. Another example is the homomorphism between groups, which maps elements in one group to corresponding elements in another group.

Is an induced map always a quotient map?

No, an induced map is not always a quotient map. While an induced map can be considered a type of quotient map, there are cases where an induced map does not satisfy the necessary properties of a quotient map. Additionally, a quotient map can be defined in ways that do not involve an induced map.

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