Topology (Part 2) - Department of Mathematics, University of Toronto
... ∴ W1U W2 is a disconnection of X . Contradiction. Corollary: If f:[a,b]→ → R, then f assumes all values between f(a) & f(b) (Intermediate Value Theorem) Proof: [a,b] is connected (in its relative topology) – this really requires proof. By above theorem, f([a,b]) (image) is connected. Suppose f(a) < ...
... ∴ W1U W2 is a disconnection of X . Contradiction. Corollary: If f:[a,b]→ → R, then f assumes all values between f(a) & f(b) (Intermediate Value Theorem) Proof: [a,b] is connected (in its relative topology) – this really requires proof. By above theorem, f([a,b]) (image) is connected. Suppose f(a) < ...
Catalogue of Useful Topological Vectorspaces
... Generally, for 0 < p ≤ ∞, define the Hardy space H p = H p (Rn ) = {u ∈ S ∗ : for some ϕ ∈ S, Mϕ u ∈ Lp } The Hardy spaces H p , with 0 < p ≤ 1 in some sense replace the Lp spaces for p < 1. The analogous definitions for p > 1 provably give Lp = H p , while this is not at all so for p ≤ 1. For p = 1 ...
... Generally, for 0 < p ≤ ∞, define the Hardy space H p = H p (Rn ) = {u ∈ S ∗ : for some ϕ ∈ S, Mϕ u ∈ Lp } The Hardy spaces H p , with 0 < p ≤ 1 in some sense replace the Lp spaces for p < 1. The analogous definitions for p > 1 provably give Lp = H p , while this is not at all so for p ≤ 1. For p = 1 ...
The Logic of Stone Spaces - New Mexico State University
... Theorem For B a complete Boolean algebra with Stone space X . 1. If B is finite, the logic of X is classical. 2. If B is infinite and atomic, the logic of X is S4.1.2. 3. Otherwise the logic of X is S4.2. Proof. Such X has a closed subspace homeomorphic to βω. We use this to build our map X −→ → Q ⊕ ...
... Theorem For B a complete Boolean algebra with Stone space X . 1. If B is finite, the logic of X is classical. 2. If B is infinite and atomic, the logic of X is S4.1.2. 3. Otherwise the logic of X is S4.2. Proof. Such X has a closed subspace homeomorphic to βω. We use this to build our map X −→ → Q ⊕ ...
A TOPOLOGY WITH ORDER, GRAPH AND AN ENUMERATION
... to the Sierpinski space. (Remember the definition of a topological manifold). The authors then have established there some properties of locally Sierpinski spaces, but left the following enumeration problem open: Let X be a finite set with n elements. Find (up to homeomorphism) the number of differe ...
... to the Sierpinski space. (Remember the definition of a topological manifold). The authors then have established there some properties of locally Sierpinski spaces, but left the following enumeration problem open: Let X be a finite set with n elements. Find (up to homeomorphism) the number of differe ...
Partitions of Unity
... 13. Corollary. (C r Urysohn’s Lemma) Let A and B be disjoint closed subsets of a C r manifold X. Then there is a C r function g : X → R such that 0 ≤ g(x) ≤ 1 for all x ∈ A, g(x) = 0 for x ∈ X and g(x) = 1 for x ∈ B. 14. Corollary. Any closed subset of a C r manifold is the zero set of a C r real-va ...
... 13. Corollary. (C r Urysohn’s Lemma) Let A and B be disjoint closed subsets of a C r manifold X. Then there is a C r function g : X → R such that 0 ≤ g(x) ≤ 1 for all x ∈ A, g(x) = 0 for x ∈ X and g(x) = 1 for x ∈ B. 14. Corollary. Any closed subset of a C r manifold is the zero set of a C r real-va ...