CONSONANCE AND TOPOLOGICAL COMPLETENESS IN
... to be lower semicontinuous (or briefly l.s.c.) if, for every open set U ⊂ X, the set {y ∈ Y : ϕ(y) ∩ U 6= ∅} is open in Y . A subset A of X which meets all values of ϕ will be called a section of ϕ. As in [3], for every collection D of subsets of X we denote by O(D) the set of open subsets of X whic ...
... to be lower semicontinuous (or briefly l.s.c.) if, for every open set U ⊂ X, the set {y ∈ Y : ϕ(y) ∩ U 6= ∅} is open in Y . A subset A of X which meets all values of ϕ will be called a section of ϕ. As in [3], for every collection D of subsets of X we denote by O(D) the set of open subsets of X whic ...
Primitive words and spectral spaces
... 1. Introduction and notations By an alphabet we mean a finite nonempty set A. The elements of A are called letters of A. A finite word over an alphabet A is a finite sequence of elements of A. The set of all finite words is denoted by A∗ . The sequence of zero letters is called the empty word and denote ...
... 1. Introduction and notations By an alphabet we mean a finite nonempty set A. The elements of A are called letters of A. A finite word over an alphabet A is a finite sequence of elements of A. The set of all finite words is denoted by A∗ . The sequence of zero letters is called the empty word and denote ...
![Forms [14 CM] and [43 W] through [43 AC] [14 CM] Kolany`s](http://s1.studyres.com/store/data/014889156_1-4ddf6cb6c42621168d150358ab1c3978-300x300.png)
Forms [14 CM] and [43 W] through [43 AC] [14 CM] Kolany`s
... that for every j ∈ J and every f ∈ Fj , dom f = Aj . Assume that for every finite J0 ⊆ J there is a function F0 such that for all j ∈ J0 , F0 | Aj ∈ Fj , then there exists a function F such that for all j ∈ J, F | Aj ∈ Fj . Kolany [1999]. [43 W] Countable products of compact Hausdorff spaces are Bai ...
... that for every j ∈ J and every f ∈ Fj , dom f = Aj . Assume that for every finite J0 ⊆ J there is a function F0 such that for all j ∈ J0 , F0 | Aj ∈ Fj , then there exists a function F such that for all j ∈ J, F | Aj ∈ Fj . Kolany [1999]. [43 W] Countable products of compact Hausdorff spaces are Bai ...
HOMEOMORPHISMS THE GROUPS OF AND
... (c) Since (X,T) is not indiscrete, there exists a proper nonempty open set A of (X,T). Let b X\A. Then X\{b} is open by (b). Thus } Is closed. Then every sngleton subset of (X,T) is closed, since (X,T) Is homogeneous by (a). Hence every fnlte subset, being a finite union of sngleton subsets is close ...
... (c) Since (X,T) is not indiscrete, there exists a proper nonempty open set A of (X,T). Let b X\A. Then X\{b} is open by (b). Thus } Is closed. Then every sngleton subset of (X,T) is closed, since (X,T) Is homogeneous by (a). Hence every fnlte subset, being a finite union of sngleton subsets is close ...
COMPACT SÍ-SOUSLIN SETS ARE Ga`S result holds for f
... Halmos [4, Theorem D, p. 221] viz. "Every compact Baire set is a ^i." Several generalizations have been established (see. for example, [1] and [7]), mostly in the direction of abstractions of Halmos' proof. For example, every Baire set is "distinguishable" and every compact distinguishable set is a ...
... Halmos [4, Theorem D, p. 221] viz. "Every compact Baire set is a ^i." Several generalizations have been established (see. for example, [1] and [7]), mostly in the direction of abstractions of Halmos' proof. For example, every Baire set is "distinguishable" and every compact distinguishable set is a ...
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.