On resolvable spaces and groups - EMIS Home
... called -bounded if for every neighborhood U of the identity there exists a subset K X with jK j such that X = K U . It is not hard to see that every subgroup of an -bounded topological group is -bounded. The metrizable @0 -bounded groups are separable, and every @0 bounded group may be em ...
... called -bounded if for every neighborhood U of the identity there exists a subset K X with jK j such that X = K U . It is not hard to see that every subgroup of an -bounded topological group is -bounded. The metrizable @0 -bounded groups are separable, and every @0 bounded group may be em ...
(1) g(S) c u,
... total order. First some examples. Let X be a totally ordered set which is a connected space in the interval topology, let £ be a subset of X containing, with t, all elements less than t, and letbe any continuous
function from X into (0, 1) whose restriction,
0O, to £ is a strictly
order-preservi ...
... total order. First some examples. Let X be a totally ordered set which is a connected space in the interval topology, let £ be a subset of X containing, with t, all elements less than t, and let
9.
... Topology is Analysis in a general setting. The concept of b-open sets was studied by Andrijevic [1]. Using the concept of b-open sets, the nets and their convergency, closure and continuity are studied. Some characterizations of b-compact spaces in terms of nets and filterbases, b-complete accumulat ...
... Topology is Analysis in a general setting. The concept of b-open sets was studied by Andrijevic [1]. Using the concept of b-open sets, the nets and their convergency, closure and continuity are studied. Some characterizations of b-compact spaces in terms of nets and filterbases, b-complete accumulat ...
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.