On c*-Compact Spaces
... containing H. Then there exists a regular open set + in such that H ∈ + and + is c*-compact subspace of . By Lemma 4.2 + is regular open in , G and by Theorem 3.1 + is c*-compact subspace of . Thus , G is locally c*-compact. Now assume , G is locally c*-compact space, so by Definition 4 ...
... containing H. Then there exists a regular open set + in such that H ∈ + and + is c*-compact subspace of . By Lemma 4.2 + is regular open in , G and by Theorem 3.1 + is c*-compact subspace of . Thus , G is locally c*-compact. Now assume , G is locally c*-compact space, so by Definition 4 ...
β1 -paracompact spaces
... α-paracompact [5] (resp., P1 -paracompact [15], S1 -paracompact [3]) spaces are defined by replacing the open cover in original definition by α-open (resp., preopen, semiopen) cover. A subset A of space X is said to be N -closed relative to X (briefly, N -closed) [9] if for S every cover {Uα : α ∈ ∆ ...
... α-paracompact [5] (resp., P1 -paracompact [15], S1 -paracompact [3]) spaces are defined by replacing the open cover in original definition by α-open (resp., preopen, semiopen) cover. A subset A of space X is said to be N -closed relative to X (briefly, N -closed) [9] if for S every cover {Uα : α ∈ ∆ ...
A GROUPOID ASSOCIATED TO DISCRETE
... ingredients: a space (phase space) X, time T, and a time evolution. In the classical case of discrete-time systems the time T is (reversible case) or (irreversible case). In the autonomous case the timeevolution law is given by an action of T on phase space (the space of all possible states of the s ...
... ingredients: a space (phase space) X, time T, and a time evolution. In the classical case of discrete-time systems the time T is (reversible case) or (irreversible case). In the autonomous case the timeevolution law is given by an action of T on phase space (the space of all possible states of the s ...
(pdf)
... An Alexandroff space is a topological space in which arbitrary intersections of open sets are open. These spaces were first introduced by P. Alexandroff in 1937 in [1] under the name of Diskrete Räume. Finite spaces are a special case of Alexandroff spaces. There is a close relationship between Ale ...
... An Alexandroff space is a topological space in which arbitrary intersections of open sets are open. These spaces were first introduced by P. Alexandroff in 1937 in [1] under the name of Diskrete Räume. Finite spaces are a special case of Alexandroff spaces. There is a close relationship between Ale ...
Relations among continuous and various non
... Let X be a topological space and p e X. X is said to have property kx at p and only if for each infinite subset A having p as an accumulation point, there is a compact subset of A + p, B, such that p e B and p is an accumulation point of B. X is a kγ space if it has property kx at each of its points ...
... Let X be a topological space and p e X. X is said to have property kx at p and only if for each infinite subset A having p as an accumulation point, there is a compact subset of A + p, B, such that p e B and p is an accumulation point of B. X is a kγ space if it has property kx at each of its points ...
Note on the Tychonoff theorem and the axiom of choice.
... We can continue transfinitely, which shows that that cardinality of P is greater than that of any other set, which is impossible. Let {Xα }α∈Λ be a collection of nonempty sets. Define a partial choice function to be a function f : I → ∪α∈I Xα where I ⊂ Λ. The set P of partial choice functions is par ...
... We can continue transfinitely, which shows that that cardinality of P is greater than that of any other set, which is impossible. Let {Xα }α∈Λ be a collection of nonempty sets. Define a partial choice function to be a function f : I → ∪α∈I Xα where I ⊂ Λ. The set P of partial choice functions is par ...
1. Introduction and preliminaries
... Proof. Let {ValaEI} be a regular semiopen cover of Y. Then there is regular open Aa such that Aac VacCl(A a) for each aEI. Sincef is almost -continuous open, f -1 (Aa) Cf -1 (V a) ef- I (Cl (Aa» eCI (f -1 (Aa»' That is, {I-I(V a) laEI} is a semiopen cover of X so {j-I(Va) U Int(Cl(f- 1(Va» Ia E I} i ...
... Proof. Let {ValaEI} be a regular semiopen cover of Y. Then there is regular open Aa such that Aac VacCl(A a) for each aEI. Sincef is almost -continuous open, f -1 (Aa) Cf -1 (V a) ef- I (Cl (Aa» eCI (f -1 (Aa»' That is, {I-I(V a) laEI} is a semiopen cover of X so {j-I(Va) U Int(Cl(f- 1(Va» Ia E I} i ...
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.