Maximal Tychonoff Spaces and Normal Isolator Covers
... is pseudometrizable. In the course of that study, disconnectedness is seen to play a major role, especially when that is of very strong in nature viz. zero-dimensionality, it is shown there that for a paracompact T2 space (X, τ ) containing no isolated points, the cardinality of such nontrivial prop ...
... is pseudometrizable. In the course of that study, disconnectedness is seen to play a major role, especially when that is of very strong in nature viz. zero-dimensionality, it is shown there that for a paracompact T2 space (X, τ ) containing no isolated points, the cardinality of such nontrivial prop ...
Part III Topological Spaces
... With this notation, the set UΛ × XA\Λ ∈ V and x ∈ UΛ × XA\Λ ⊂ V ⊂ W. This shows that every open set in X may be written as a union of elements from V, i.e. V is a base for the product topology. Notation 10.26 Let Ei ⊂ 2Xi be a collection of subsets of a set Xi for each i = 1, 2, . . . , n. We will w ...
... With this notation, the set UΛ × XA\Λ ∈ V and x ∈ UΛ × XA\Λ ⊂ V ⊂ W. This shows that every open set in X may be written as a union of elements from V, i.e. V is a base for the product topology. Notation 10.26 Let Ei ⊂ 2Xi be a collection of subsets of a set Xi for each i = 1, 2, . . . , n. We will w ...
FUNDAMENTAL GROUPS OF TOPOLOGICAL STACKS
... see Theorem 9.1, Theorem 10.4, and Remark 10.7 – Corollary 10.5 should also be of interest. For the above results to hold, one needs a technical hypothesis which is called the slice condition (Definitions 5.1 and 5.4). As the terminology suggests, this notion is modeled on the slice property of comp ...
... see Theorem 9.1, Theorem 10.4, and Remark 10.7 – Corollary 10.5 should also be of interest. For the above results to hold, one needs a technical hypothesis which is called the slice condition (Definitions 5.1 and 5.4). As the terminology suggests, this notion is modeled on the slice property of comp ...
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.