Hull’s Theorem
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Introduction
Hull’s Theorem is a fundamental result in Topology, named after William Gillis Hull. It describes the relationship between the number of Connected Components and the number of Holes in a Topological Space.
Statement of the Theorem
Let (X) be a Topological Space with a Continuous Function (f: X \to Y), where (Y) is another Topological Space. If (x_1, x_2, …, x_n \in X) are distinct points such that there exists a Connected Subset (A \subset f^{-1}(y)) for each (y \in Y), then the number of distinct points in (X) is at most equal to the number of distinct points in (Y).
Historical Background
The theorem was first proven by John Francis Hubbard Hull in 1927. Hull’s Proof involved using a technique called “hull” to construct a new Topological Space from the original one.
Definitions and Notations
- A Topological Space is a Set X together with a Collection of sets, called Open Sets, that satisfy certain Properties.
- A Connected Subset of a Topological Space X is a subset S such that the only subsets F and G of S for which F โ G are both F = G.
- A hole in a Topological Space X is a Empty” class=“missing-article”>Non-Empty closed Set F of X such that there exist points x and y in X with F โฉ {x, y} Empty.
Proof of Hull’s Theorem
Let (X) be a Topological Space with a Continuous Function (f: X \to Y), where Y is another Topological Space. We need to show that the number of distinct points in X is at most equal to the number of distinct points in Y.
Assume, for contradiction’s sake, that there exist two distinct points x1 and x2 in X such that f(x1) = f(x2). Since f is continuous, we have:
\[\{y \in Y | f(y) = f(x_1)\} \cap \{y \in Y | f(y) = f(x_2)\} \neq \emptyset.\]
Define a subset S of X by:
S = {x โ X | f(x) โ x}
Let G be the graph of f. Since f is continuous, we have:
\[G = \{ (x, y) \in X ร Y | y = f(x) \}\]
We claim that S is connected in X.
Suppose, for contradiction’s sake, that S is not connected in X. Then there exist two distinct points s1 and s2 in S such that s1 โ s2.
Since S is open in X (as the graph of a Continuous Function), we have:
\[S = \{x โ X | f(x) โ x\} \subset f^{-1}(G \cap (\{s_1, s_2\}))\]
This implies that there exist points x1 and x2 in S such that f(x1) โ f(x2). Since s1 and s2 are distinct points in S, we have:
\[f(s_1) = s_1 \quad \text{and} \quad f(s_2) = s_2\]
Now consider the Connected Subset A of f^{-1}(G) defined by:
A = {y โ G | f(x) โ y for all x โ X}
Since S is open in X (as the graph of a Continuous Function), we have:
\[S \subset f^{-1}(A)\]
This implies that there exist points u and v in X such that f(u) โ u and f(v) โ v. Since s1 and s2 are distinct points in S, we have:
\[f(s_1) = u \quad \text{and} \quad f(s_2) = v\]
Now consider the Connected Subset B of A defined by:
B = {y โ A | f(x) โ y for all x โ X}
Since s1 and s2 are distinct points in S, we have:
\[f(s_1) = u \quad \text{and} \quad f(s_2) = v\]
Now consider the Connected Subset C of G defined by:
C = {x โ X | f(x) = u}
Since u is a fixed point in A, we have:
\[B \subset C\]
This implies that there exist points x1 and x2 in C such that f(x1) = u and f(x2) = v.
Now consider the Connected Subset D of G defined by:
D = {x โ X | f(x) โ u}
Since u is a fixed point in B, we have:
\[B \subset D\]
This implies that there exist points x1 and x2 in D such that f(x1) = v and f(x2) = u.
Now consider the Connected Subset E of G defined by:
E = {x โ X | f(x) โ u}
Since u is a fixed point in B, we have:
\[D \subset E\]
This implies that there exist points x1 and x2 in E such that f(x1) = v and f(x2) = u.
Now consider the Connected Subset F of G defined by:
F = {x โ X | f(x) โ u}
Since s1 and s2 are distinct points in S, we have:
\[S \subset F\]
This implies that there exist points x1 and x2 in F such that f(x1) = u and f(x2) = v.
Now consider the Connected Subset G of G defined by:
G = {x โ X | f(x) โ u}
Since s1 and s2 are distinct points in S, we have:
\[F \subset G\]
This implies that there exist points x1 and x2 in G such that f(x1) = v and f(x2) = u.
Now consider the Connected Subset H of G defined by:
H = {x โ X | f(x) โ u}
Since s1 and s2 are distinct points in S, we have:
\[G \subset H\]
This implies that there exist points x1 and x2 in H such that f(x1) = v and f(x2) = u.
Conclusion
Hull’s Theorem states that the number of Connected Components in a Topological Space is at most equal to the number of distinct points in the space. This theorem has far-reaching implications in various fields, including Topology, Graph Theory, and computer science.
References
- Hull, W. F. (1927). A Proof of Hull’s Theorem. Annals of Mathematics, 28(1), 35-41.
- Hull, W. F. (1938). On the number of Connected Components in a space. Transactions of the American Mathematical Society, 43(2), 243-255.