Prove 1. If X is any space, A is a closed subset of X, and p is not in X, the space X +¢{p}...

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Prove 1. If X is any space, A is a closed subset of X, and p is not in X, the space X +¢{p} resulting from the function f which takes A to {p}is homeomorphic to the quotient space of X obtained by identifying A to a single point. Refence XXXXXXXXXXDefinition. Let X, be a topological space, for each a € A, and let X* = {(x, «| xe X,}, with the topology being defined on X? in the obvious way, to make it homeomorphic to X,. The collection of spaces X* is different from the collection of spaces X,, then, only in that X} 0 Xf = oifa # B. Now define a topology on X = |),., X? as follows: U 
	
Document Preview: AttachingsProve1. If X is any space, A is a closed subset of X, and p is not in X, the space X +f {p} resulting from the function f which takes A to {p} is homeomorphic to the quotient space of X obtainedby identifying A to a single point.Refence 1

Robert · Accepted Answer

We have inclusion map i from i : X → X
⊔
{p} which is continuous. Proof
is trivial which is as follows:
If U is open set in X
⊔
{p} by definition of topology of X
⊔
{p} we know that
U ∩X is open in X. But U ∩X = i−1(U). Hence i is continuous.
Now Let R1 be relation on X which identifies A to single point. And let R2
be relation on X
⊔
{p} which defines X +f {p}.
Define F which makes following diagram commutative.
X
X
⊔
{p}
X +f {p}X/R1
i
q2
q1
F
That is F is defined as
F ◦ q1 = q2 ◦ i
Explicitly F is defined as
F (x) =
{
[x] if x /∈ A;
[p] if x ∈ A.

Prove 1. If X is any space, A is a closed subset of X, and p is not in X, the space X +¢{p} resulting from the function f which takes A to {p}is homeomorphic to the quotient space of X obtained by...

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