class notes - Math User Home Pages
... ‚ The set tr´8, ´M q | M P Ru is a nbd base at ´8 in R˚ . This nbd base is called the standard nbd base at ´8. ‚ The set tr´8, ´M q | M P Nu is a countable nbd base at ´8 in R˚ . This nbd base is called the standard countable nbd base at ´8. ‚ Let pX, dq, for every a P X, Then tBpa, rq | r ą 0u is a ...
... ‚ The set tr´8, ´M q | M P Ru is a nbd base at ´8 in R˚ . This nbd base is called the standard nbd base at ´8. ‚ The set tr´8, ´M q | M P Nu is a countable nbd base at ´8 in R˚ . This nbd base is called the standard countable nbd base at ´8. ‚ Let pX, dq, for every a P X, Then tBpa, rq | r ą 0u is a ...
ON MAXIMAL, MINIMAL OPEN AND CLOSED SETS 1. Introduction
... Theorem 3.2 (Nakaoka and Oda [3]). If E is a maximal closed set and F is any closed set, then either E ∪ F = X or F ⊂ E. Definition 3.3 (Nakaoka and Oda [3]). A proper nonempty closed subset E of X is said to be a minimal closed set if any closed set which is contained in E is E or ∅. Theorem 3.4 (N ...
... Theorem 3.2 (Nakaoka and Oda [3]). If E is a maximal closed set and F is any closed set, then either E ∪ F = X or F ⊂ E. Definition 3.3 (Nakaoka and Oda [3]). A proper nonempty closed subset E of X is said to be a minimal closed set if any closed set which is contained in E is E or ∅. Theorem 3.4 (N ...
Non-commutative Donaldson--Thomas theory and vertex operators
... core A of the t-structure of the derived category. In Section 2, the definition of Euler characteristic version of open non-commutative Donaldson–Thomas invariants is provided. Then, we compute the generating function using vertex operators in Section 3. Finally, we study open Donaldson–Thomas inva ...
... core A of the t-structure of the derived category. In Section 2, the definition of Euler characteristic version of open non-commutative Donaldson–Thomas invariants is provided. Then, we compute the generating function using vertex operators in Section 3. Finally, we study open Donaldson–Thomas inva ...
Universal nowhere dense and meager sets in Menger manifolds
... standard technique of skeletoids cannot be applied to constructing K-universal or σK-universal sets for families K ⊂ Z. In [2] and [3] the authors using the technique of tame open set and tame Gδ -set, constructed Z0 -universal and σZ0 -universal sets in manifolds modeled on finite-dimensional and i ...
... standard technique of skeletoids cannot be applied to constructing K-universal or σK-universal sets for families K ⊂ Z. In [2] and [3] the authors using the technique of tame open set and tame Gδ -set, constructed Z0 -universal and σZ0 -universal sets in manifolds modeled on finite-dimensional and i ...
Selection principles and countable dimension
... Marion Scheepers Department of Mathematics, Boise State University, ...
... Marion Scheepers Department of Mathematics, Boise State University, ...
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.