Glueing together normal topological spaces at a closed subset

In summary, what the person is trying to ask is how to prove that a space is normal. The person goes on to say that they would rather not use abstract non-sense, so they can understand the proof better.
  • #1
conquest
133
4
Hi all!

My question is the following. Suppose we have two normal topological spaces X and Y and we have a continuous map from a closed subset A of X to Y. Then we can construct another topological space by "glueing together" X and Y at A and f(A). By taking the quotient space of the disjoint union of X and Y by the equivalence relation that

x is equivalent to y if:

1) x=y
2) x,y are elements of A and f(x)=f(y)
or
3) x is an element of A and y is an element of Y and f(x)=y
or x is an element of Y and y is an element of A and x=f(y).

My question is how can you prove that this constructed space is again normal?
 
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  • #2
The space obtained from this construction is called an adjunction space and is typically denoted by ##X \cup_f Y##. It forms a pushout in the category of topological spaces. With this and the normality of X and Y in mind, it's not too difficult to show that you can separate closed sets in ##X \cup_f Y## by a continuous function.
 
  • #3
Okay I believe this, but I would rather find a way to prove it without using any abstract non-sense so that I have an idea of where it is coming from.

So basically the question is if I have non-intersecting closed sets in this adjunction I look at the pre-image in the disjoint union of X and Y (where they are again closed and non-intersecting). Since I know X and Y are normal I can construct open sets now in both X and Y that don't intersect and contain the earlier closed sets (i.e. definition of normal).
but how then do I make sure that when I project them onto the adjunction they are again open and non-intersecting?
 
  • #4
The abstract nonsense here is just a reformulation of what it means to give a space the quotient topology, so it's really not so abstract.

It's probably best not to try to separate by open sets, but to separate with continuous functions, like I mentioned in my post. This is where the abstract nonsense (which again is not so abstract) will be helpful.
 
  • #5
aaah thank you!

Suddenly it all makes sense thanks!
 

What does it mean to "glue together" topological spaces?

When we say "glue together" in the context of topological spaces, we mean to take two separate spaces and identify certain points on their boundaries to create a new, larger space. This is often done to create new topological spaces that have desirable properties.

What is a "closed subset" in this context?

A closed subset is a subset of a topological space that contains all of its boundary points. In other words, it is a subset that "touches" all the points on the edge of the space. When we glue together topological spaces at a closed subset, we are essentially connecting the boundaries of these spaces.

What is the resulting topological space after gluing together at a closed subset?

The resulting topological space is called the "quotient space." It is a new topological space that is created by identifying points on the boundaries of the original spaces. This new space inherits properties from the original spaces, but may also have its own unique properties.

What are some applications of gluing together topological spaces at a closed subset?

One application is in topology itself, where this technique is used to create new spaces for studying and understanding mathematical properties. It can also be used in other fields, such as in physics, where spaces with specific properties are needed for modeling certain phenomena.

Are there any limitations to gluing together topological spaces at a closed subset?

Yes, there are limitations. For example, the resulting quotient space may not always be a topological space itself. In addition, the choice of which points to identify on the boundaries can greatly affect the properties of the resulting space, so careful consideration is needed when using this technique.

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