Rare α-Continuity
... each x ∈ X and each open set G ⊂ Y containing f (x), there exist a rare set RG with G ∩ Cl(RG ) = ∅ and U ∈ α(X, x) (resp. an open set U ⊂ X containing x, U ∈ P O(X, x) and U ∈ SO(X, x)) such that f (U ) ⊂ G ∪ RG . Example 2.1. Let X = {a, b, c}, τ = {X, ∅, {c}} and σ = {X, ∅, {a}, {b}, {a, b}}. Let ...
... each x ∈ X and each open set G ⊂ Y containing f (x), there exist a rare set RG with G ∩ Cl(RG ) = ∅ and U ∈ α(X, x) (resp. an open set U ⊂ X containing x, U ∈ P O(X, x) and U ∈ SO(X, x)) such that f (U ) ⊂ G ∪ RG . Example 2.1. Let X = {a, b, c}, τ = {X, ∅, {c}} and σ = {X, ∅, {a}, {b}, {a, b}}. Let ...
Notes on Topological Dimension Theory
... The definition of Čech homology requires the notion of inverse limit; special cases of this concept appear in Hatcher, but since we need the general case we must begin from scratch. Definition. A codirected set is a pair (A, ≺) consisting of a set A and a binary operation ≺ such that the following ...
... The definition of Čech homology requires the notion of inverse limit; special cases of this concept appear in Hatcher, but since we need the general case we must begin from scratch. Definition. A codirected set is a pair (A, ≺) consisting of a set A and a binary operation ≺ such that the following ...
Partial Metric Spaces - Department of Computer Science
... Then (X, p) is a partial metric space, and p(x, x) = |x|. Conversely, if (X, p) is a partial metric space, then (X, d p , | · |), where (as before) d p (x, y) = 2 p(x, y) − p(x, x) − p(y, y) and |x| = p(x, x), is a weighted metric space. It can be seen that from either space we can move to the other ...
... Then (X, p) is a partial metric space, and p(x, x) = |x|. Conversely, if (X, p) is a partial metric space, then (X, d p , | · |), where (as before) d p (x, y) = 2 p(x, y) − p(x, x) − p(y, y) and |x| = p(x, x), is a weighted metric space. It can be seen that from either space we can move to the other ...
A TOPOLOGICAL CONSTRUCTION OF CANONICAL EXTENSIONS
... We will be interested in special sorts of saturated sets: compact saturated sets, open sets and filters. In that light, we define • K(X): the collection of compact saturated subsets of X; • O(X): the collection of open subsets of X; and • F(X): the collection of filters of X. Intersections of these ...
... We will be interested in special sorts of saturated sets: compact saturated sets, open sets and filters. In that light, we define • K(X): the collection of compact saturated subsets of X; • O(X): the collection of open subsets of X; and • F(X): the collection of filters of X. Intersections of these ...
Open subgroups and Pontryagin duality
... P r o o f The discrete group G/A admits sufficiently many continuous'characters, which proves (a). For (b), see [4, Lemma 3.3]. Statement (c) follows from (1.2). Now, (b) says that # " G ---,A^ is surjective. To prove that #~ is open, take an arbitrary compact subset X of G. In view of (1.1), we onl ...
... P r o o f The discrete group G/A admits sufficiently many continuous'characters, which proves (a). For (b), see [4, Lemma 3.3]. Statement (c) follows from (1.2). Now, (b) says that # " G ---,A^ is surjective. To prove that #~ is open, take an arbitrary compact subset X of G. In view of (1.1), we onl ...
6. “× º - 筑波学院大学
... ∪ aU(z(α)) : α < κ s is an Fσ -subset of X. It follows from Lemma 1 that Z is an Fσ -subset of X. We need the following two lemmas to show Theorem 3. Lemma 2. Let Y be a GO-space. If there exists a dense subspace X of Y such that the following condition is satisfied, then Y is first-countable : Ever ...
... ∪ aU(z(α)) : α < κ s is an Fσ -subset of X. It follows from Lemma 1 that Z is an Fσ -subset of X. We need the following two lemmas to show Theorem 3. Lemma 2. Let Y be a GO-space. If there exists a dense subspace X of Y such that the following condition is satisfied, then Y is first-countable : Ever ...
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.