Redalyc.On a class of ay-open sets in a topological space
... A = {a}, since the only αγ-open supersets of A are {a, c} and X, then A is αγ-g.closed. But it is easy to see that A is not αγ-closed. Theorem 2.29. A subset A of (X, τ) is αγ-g.closed if and only if αγCl({x}) ∩ A ≠ φ, holds for every x ∈ αγCl(A). Proof. Let U be an αγ-open set such that A ⊆ U and l ...
... A = {a}, since the only αγ-open supersets of A are {a, c} and X, then A is αγ-g.closed. But it is easy to see that A is not αγ-closed. Theorem 2.29. A subset A of (X, τ) is αγ-g.closed if and only if αγCl({x}) ∩ A ≠ φ, holds for every x ∈ αγCl(A). Proof. Let U be an αγ-open set such that A ⊆ U and l ...
The Hurewicz covering property and slaloms in the Baire space
... Proof. (1)⇒(2). Assume that g ∈ N% N bounds Y . Define inductively h ∈ N% N by h(0) = g(0), h(n + 1) = g(h(n)) + 1. Then for each f ∈ Y and all but finitely many n, h(n) ≤ f (h(n)) ≤ g(h(n)) < h(n + 1), that is, f (h(n)) ∈ [h(n), h(n + 1)). (2)⇒(1). Assume that Y admits a slalom g. Let h be a functi ...
... Proof. (1)⇒(2). Assume that g ∈ N% N bounds Y . Define inductively h ∈ N% N by h(0) = g(0), h(n + 1) = g(h(n)) + 1. Then for each f ∈ Y and all but finitely many n, h(n) ≤ f (h(n)) ≤ g(h(n)) < h(n + 1), that is, f (h(n)) ∈ [h(n), h(n + 1)). (2)⇒(1). Assume that Y admits a slalom g. Let h be a functi ...
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.