Math 446–646 Important facts about Topological Spaces
... (c) Hausdorff property or property T2 : A topological space (X, T ) has the Hausdorff property if given two different point x, y ∈ X, there exist open sets O1 , O2 ∈ T such that x ∈ O1 , y ∈ O2 and O1 ∩ O2 = ∅. Any space with the property T2 also has property T1 , but the converse is not true. If (X ...
... (c) Hausdorff property or property T2 : A topological space (X, T ) has the Hausdorff property if given two different point x, y ∈ X, there exist open sets O1 , O2 ∈ T such that x ∈ O1 , y ∈ O2 and O1 ∩ O2 = ∅. Any space with the property T2 also has property T1 , but the converse is not true. If (X ...
Course Overview
... (1) If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the greater angle is opposite the longer side. (2) If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the greater angle. ...
... (1) If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the greater angle is opposite the longer side. (2) If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the greater angle. ...
Topological vector spaces - SISSA People Personal Home Pages
... vector spaces it is also sufficient. Theorem 5.1. If X is a topological vector space with a countable base, then there is a metric d such that (1) d is compatible with τ ; (2) the open balls of 0 are balanced; (3) d is translation invariant, i.e. d(x + z, y + z) = d(x, y). If X is locally convex, th ...
... vector spaces it is also sufficient. Theorem 5.1. If X is a topological vector space with a countable base, then there is a metric d such that (1) d is compatible with τ ; (2) the open balls of 0 are balanced; (3) d is translation invariant, i.e. d(x + z, y + z) = d(x, y). If X is locally convex, th ...
Examples of topological spaces
... Example 12. Let A = {1, 2, 3} with the topology τ = {∅, {1}, {1, 2}, A}. Then the constant sequence 1, 1, 1, 1, . . . converges to 1; it also converges to 2 and to 3. Observe that A is not T1 : there is no open set around 1 separating it from 2. Example 13. Consider Z with the cofinite topology. For ...
... Example 12. Let A = {1, 2, 3} with the topology τ = {∅, {1}, {1, 2}, A}. Then the constant sequence 1, 1, 1, 1, . . . converges to 1; it also converges to 2 and to 3. Observe that A is not T1 : there is no open set around 1 separating it from 2. Example 13. Consider Z with the cofinite topology. For ...
On upper and lower contra-ω-continuous multifunctions
... Int(A) denote the closure of A with respect to τ and the interior of A with respect to τ , respectively. Recently, as generalization of closed sets, the notion of ω-closed sets were introduced and studied by Hdeib [8]. A point x ∈ X is called a condensation point of A if for each U ∈ τ with x ∈ U , ...
... Int(A) denote the closure of A with respect to τ and the interior of A with respect to τ , respectively. Recently, as generalization of closed sets, the notion of ω-closed sets were introduced and studied by Hdeib [8]. A point x ∈ X is called a condensation point of A if for each U ∈ τ with x ∈ U , ...
Compact Orthoalgebras - Susquehanna University
... jointly continuous, the relation P ⊥ Q iff P Q = QP = 0 is closed. Since addition and subtraction are continuous, the partial operation P, Q 7→ P ⊕ Q := P + Q is continuous on ⊥, as is the operation P 7→ P 0 := 1 − P . Thus, L(H) is a latticeordered topological orthoalgebra. It is not, however, a to ...
... jointly continuous, the relation P ⊥ Q iff P Q = QP = 0 is closed. Since addition and subtraction are continuous, the partial operation P, Q 7→ P ⊕ Q := P + Q is continuous on ⊥, as is the operation P 7→ P 0 := 1 − P . Thus, L(H) is a latticeordered topological orthoalgebra. It is not, however, a to ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.