Lecture 02 - UWO Math Dept
... weak Hausdorff spaces. A space X is compactly generated if a subset Z is closed if and only if Z ∩ K is closed for all maps K → X with K compact. A compactly generated space X is weakly Hausdorff if and only if the image of the diagonal ∆ : X → X × X is closed in X × X, where the product is in the c ...
... weak Hausdorff spaces. A space X is compactly generated if a subset Z is closed if and only if Z ∩ K is closed for all maps K → X with K compact. A compactly generated space X is weakly Hausdorff if and only if the image of the diagonal ∆ : X → X × X is closed in X × X, where the product is in the c ...
![arXiv:math/0201251v1 [math.DS] 25 Jan 2002](http://s1.studyres.com/store/data/014913155_1-788f445ae3a0c155aa768d6df676e28b-300x300.png)
arXiv:math/0201251v1 [math.DS] 25 Jan 2002
... is a metric equivalent to d such that h is an isometry of (X, d∗ ). Suppose x ∈ X. Let Y = {hn (x)}. To prove Y is locally d∗ complete suppose y ∈ Y. Choose γ > 0 so that A = Bd (y, γ) ∩ {hn (x)} is d complete. Choose β > 0 so that d(y, z) < β ⇒ d∗ (y, z) < γ. Let C = Bd∗ (y, β) ∩ {hn (x)}. Note C ⊂ ...
... is a metric equivalent to d such that h is an isometry of (X, d∗ ). Suppose x ∈ X. Let Y = {hn (x)}. To prove Y is locally d∗ complete suppose y ∈ Y. Choose γ > 0 so that A = Bd (y, γ) ∩ {hn (x)} is d complete. Choose β > 0 so that d(y, z) < β ⇒ d∗ (y, z) < γ. Let C = Bd∗ (y, β) ∩ {hn (x)}. Note C ⊂ ...
Weakly b-Open Functions
... Hence, f (U ) and f (V ) are b-open in Y . Thus, Y has been decomposed into two non-empty disjoint b-open sets. This is contrary to the hypothesis that Y is a b-connected space. Thus, X is connected. Recall that a subset K of a space X is said to be quasi H-closed [15] ( resp. b-compact [7] ) relati ...
... Hence, f (U ) and f (V ) are b-open in Y . Thus, Y has been decomposed into two non-empty disjoint b-open sets. This is contrary to the hypothesis that Y is a b-connected space. Thus, X is connected. Recall that a subset K of a space X is said to be quasi H-closed [15] ( resp. b-compact [7] ) relati ...
General Topology
... in the form A ∋ x. So, the origin of notation is sort of ignored, but a more meaningful similarity to the inequality symbols < and > is emphasized. To state that x is not an element of A, we write x 6∈ A or A 6∋ x. § 1 ◦ 2 Equality of Sets A set is determined by its elements. It is nothing but a col ...
... in the form A ∋ x. So, the origin of notation is sort of ignored, but a more meaningful similarity to the inequality symbols < and > is emphasized. To state that x is not an element of A, we write x 6∈ A or A 6∋ x. § 1 ◦ 2 Equality of Sets A set is determined by its elements. It is nothing but a col ...
Ordered Quotients and the Semilattice of Ordered
... constructive, giving a method for defining an ordered decomposition space, and, consequently, an ordered quotient space. Also presented is a definition for an ordered quotient map which fits neatly into the rest of the theory. We begin by reviewing pertinent notions from the theory of ordered topologi ...
... constructive, giving a method for defining an ordered decomposition space, and, consequently, an ordered quotient space. Also presented is a definition for an ordered quotient map which fits neatly into the rest of the theory. We begin by reviewing pertinent notions from the theory of ordered topologi ...
CLASSIFYING THE TYPES OF PRINCIPAL GROUPOID C
... We can weaken the conditions in Proposition 3.8 and show that, for principal groupoids, ψ is a homeomorphism when G0 /G is a T0 space. In doing this, we actually describe the ideal structure of the associated groupoid C ∗ -algebra. We will also prove a generalization of Gootman’s Theorem 1.1 for pri ...
... We can weaken the conditions in Proposition 3.8 and show that, for principal groupoids, ψ is a homeomorphism when G0 /G is a T0 space. In doing this, we actually describe the ideal structure of the associated groupoid C ∗ -algebra. We will also prove a generalization of Gootman’s Theorem 1.1 for pri ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.