Continuous Maps and Hausdorff Spaces on [0, 1] x {0, 1}

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In summary: X × {1}, it follows that K and L are compact. Finally, since A is a subset of X, it follows that K ∩ L is homeomorphic to A. In summary, we have shown that for a topological space X and a subset A of X, the quotient space [X × {0, 1}] /R is Hausdorff if and only if X is Hausdorff and A is a closed subset of X. Furthermore, for X = [0, 1] and any subset A of [0, 1], the quotient space [X × {0, 1}] /R is compact and the subsets K = p(X × {0}) and L = p
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kazuyak
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Let X be a topological space and A a subset of X . On X × {0, 1} define the
partition composed of the pairs {(a, 0), (a, 1)} for a ∈ A , and of the singletons {(x, i)} if x ∈ X \ A and i ∈ {0, 1}.
Let R be the equivalence relation defined by this partition, let Y be the quotient space
[X × {0, 1}] /R and let p : X × {0, 1} → Y be the quotient map.

(1) Prove that there exists a continuous map f : Y → X such that f ◦ p(x, i) = x for every x ∈ X and i ∈ {0, 1} .
Prove that Y is Hausdorff if and only if X is Hausdorff and A is a closed subset of X .

(2) Consider the above construction for X = [0, 1] and A an arbitrary subset of [0, 1].
Prove that Y is compact. Prove that K = p(X × {0}) and L = p(X × {1}) are compact, and that K ∩ L is homeomorphic to A .

Any input is appreciated!
 
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Thanks. (1) For the first part, let f : Y → X be defined by f([x, i]) = x. It is easy to see that f ◦ p(x, i) = x for every x ∈ X and i ∈ {0, 1}. To prove that f is continuous, let U be an open set in X. Then f^-1(U) = p(U x {0}) U p(U x {1}), which is open in Y as a union of open sets in Y. Hence f is continuous. For the second part, suppose X is Hausdorff. Let [x, 0] and [x, 1] be two distinct points in Y. Then p^-1([x, 0]) = {x} x {0} and p^-1([x, 1]) = {x} x {1} are disjoint open sets in X x {0, 1}, so there exist disjoint open sets U and V in X such that x ∈ U and x ∈ V. Then U and V are disjoint open sets in Y containing [x, 0] and [x, 1], respectively, so Y is Hausdorff. Conversely, suppose Y is Hausdorff. Let x, y ∈ X be distinct points. Then [x, 0] and [y, 1] are distinct points in Y, so there exist disjoint open sets U and V in Y containing [x, 0] and [y, 1], respectively. Then p^-1(U) and p^-1(V) are disjoint open sets in X x {0, 1} containing (x, 0) and (y, 1), respectively, so there exist disjoint open sets U' and V' in X such that x ∈ U' and y ∈ V'. Hence X is Hausdorff. (2) Since X = [0, 1] is compact, it follows that X × {0, 1} is compact. Now, since R is an equivalence relation, it follows that Y is compact, being the quotient of a compact space. Since K ⊂ X × {0} and L
 

1. What is a continuous map?

A continuous map is a function between two topological spaces that preserves the topological structure. In other words, small changes in the input result in small changes in the output. In the context of [0,1] x {0,1}, a continuous map would be a function that maps points on the square to points on the line segment in a way that preserves the topological properties of both spaces.

2. What is a Hausdorff space?

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighborhoods. In simpler terms, this means that points in a Hausdorff space can be separated by open sets. In the context of [0,1] x {0,1}, this means that any two distinct points on the square can be separated by open sets, and similarly for the line segment.

3. How do continuous maps and Hausdorff spaces relate to each other?

In general, continuous maps between Hausdorff spaces are also continuous maps between topological spaces. In the specific case of [0,1] x {0,1}, since both the square and the line segment are Hausdorff spaces, any continuous map between them would also be a continuous map between topological spaces. This means that the topological structure of both spaces is preserved by the mapping.

4. Can [0,1] x {0,1} be a Hausdorff space?

Yes, [0,1] x {0,1} can be a Hausdorff space. This is because both [0,1] and {0,1} are Hausdorff spaces, and the product of two Hausdorff spaces is also a Hausdorff space.

5. What are some applications of continuous maps and Hausdorff spaces on [0,1] x {0,1}?

Continuous maps and Hausdorff spaces have many applications in mathematics and other fields. In the specific context of [0,1] x {0,1}, they can be used to model various physical systems, such as the movement of particles on a surface or the flow of fluids through a porous medium. They can also be applied in computer science for tasks such as image compression or data analysis. Additionally, the study of continuous maps and Hausdorff spaces on [0,1] x {0,1} can lead to a better understanding of topological structures and their properties.

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