On Upper and Lower Weakly c-e-Continuous Multifunctions 1
... (i) lower weakly continuous [13] if for each x ∈ X and each open set V of Y such that x ∈ F − (V ), there exists an open set U containing x such that U ⊂ F − ((V )); (ii) upper weakly continuous [13] for each x ∈ X and each open set V of Y such that x ∈ F + (V ), there exists an open set U containin ...
... (i) lower weakly continuous [13] if for each x ∈ X and each open set V of Y such that x ∈ F − (V ), there exists an open set U containing x such that U ⊂ F − ((V )); (ii) upper weakly continuous [13] for each x ∈ X and each open set V of Y such that x ∈ F + (V ), there exists an open set U containin ...
STABLE TOPOLOGICAL CYCLIC HOMOLOGY IS TOPOLOGICAL
... G-universe’ U . Then t(L; P )(V ) = THH(L; P ; S V ), with the obvious G-maps σ: S W −V ∧ t(L; P )(V ) → t(L; P )(W ) as prespectrum structure maps. Here S V is the one-point compactification of V and W − V is the orthogonal complement of V in W . We also define a G-spectrum T (L; P ) associated wit ...
... G-universe’ U . Then t(L; P )(V ) = THH(L; P ; S V ), with the obvious G-maps σ: S W −V ∧ t(L; P )(V ) → t(L; P )(W ) as prespectrum structure maps. Here S V is the one-point compactification of V and W − V is the orthogonal complement of V in W . We also define a G-spectrum T (L; P ) associated wit ...
Some results on sequentially compact extensions
... consisting of Tychonoff sequential compactifications of X. (As we have already remarked, those sets can be empty.) In the first section of this paper we observe that, for Tychonoff LSC spaces, oX can fail to be Tychonoff. However, we prove that every Tychonoff LSC space has a Tychonoff one-point seq ...
... consisting of Tychonoff sequential compactifications of X. (As we have already remarked, those sets can be empty.) In the first section of this paper we observe that, for Tychonoff LSC spaces, oX can fail to be Tychonoff. However, we prove that every Tychonoff LSC space has a Tychonoff one-point seq ...
ON WEAK FORMS OF PREOPEN AND PRECLOSED
... functions. In 1985, D. A. Rose [26] and D. A. Rose with D. S. Janckovic [27] have defined the notions of weakly open and weakly closed functions in topology respectively. This paper is devoted to present the class of weakly preopen functions (resp. weakly preclosed functions) as a new generalization ...
... functions. In 1985, D. A. Rose [26] and D. A. Rose with D. S. Janckovic [27] have defined the notions of weakly open and weakly closed functions in topology respectively. This paper is devoted to present the class of weakly preopen functions (resp. weakly preclosed functions) as a new generalization ...
Real Analysis
... Let {Xλ }λ∈Λ be a collection of compact Q topological spaces indexed by a set Λ. Then the Cartesian product λ∈Λ Xλ , with the product topology, also is compact. Proof (continued). But B (by choice) has the finite intersection property and contains the collection F. Let F be a set in F. Then F is in ...
... Let {Xλ }λ∈Λ be a collection of compact Q topological spaces indexed by a set Λ. Then the Cartesian product λ∈Λ Xλ , with the product topology, also is compact. Proof (continued). But B (by choice) has the finite intersection property and contains the collection F. Let F be a set in F. Then F is in ...
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.