Metric and Topological Spaces T. W. K¨orner October 16, 2014
... We call B(x, r) the open ball with centre x and radius r. The following result is very important for the course, but is also very easy to check. Theorem 4.7. If (X, d) is a metric space, then the following statements are true. (i) The empty set ∅ and the space X S are open. (ii) If Uα is open for al ...
... We call B(x, r) the open ball with centre x and radius r. The following result is very important for the course, but is also very easy to check. Theorem 4.7. If (X, d) is a metric space, then the following statements are true. (i) The empty set ∅ and the space X S are open. (ii) If Uα is open for al ...
Recent Progress in the Theory of Generalized Closed Sets ∗
... Definition 3. A topological space (X, τ ) is called sg–compact if every cover by sg–open sets has a finite subcover. The class of sg–compact spaces has been introduced by Caldas [3], Devi, Balachandran and Maki [9] and Tapi, Thakur and Sonwalkar [27]. Sg–compact spaces are quite interesting because ...
... Definition 3. A topological space (X, τ ) is called sg–compact if every cover by sg–open sets has a finite subcover. The class of sg–compact spaces has been introduced by Caldas [3], Devi, Balachandran and Maki [9] and Tapi, Thakur and Sonwalkar [27]. Sg–compact spaces are quite interesting because ...
COUNTABLE DENSE HOMOGENEITY AND THE DOUBLE ARROW
... consider the equivalence relation ∼ on T obtained by defining x ∼ y if either x = y or [min {x, y}, max {x, y}] is a jump in T . Since T is crowded, each equivalence class of ∼ consists of at most two points. Since the equivalence classes are convex, the quotient T / ∼ can be linearly ordered in a na ...
... consider the equivalence relation ∼ on T obtained by defining x ∼ y if either x = y or [min {x, y}, max {x, y}] is a jump in T . Since T is crowded, each equivalence class of ∼ consists of at most two points. Since the equivalence classes are convex, the quotient T / ∼ can be linearly ordered in a na ...
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... Let V be a finite dimensional vector space (over some field) with dimension n. Let P G(V ) be its lattice of subspaces, also known as the projective geometry of V . It is well-known that we can associate each element a ∈ P G(V ) a unique integer dim(a), namely, the dimension of the a as a subspace o ...
... Let V be a finite dimensional vector space (over some field) with dimension n. Let P G(V ) be its lattice of subspaces, also known as the projective geometry of V . It is well-known that we can associate each element a ∈ P G(V ) a unique integer dim(a), namely, the dimension of the a as a subspace o ...
ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION
... We recall that a topological space X is spectral whenever X is sober and the compact and open subsets are closed under finite intersections and form a base for the topology of X. Note that in particular every spectral space is compact. A continuous map f : X → Y between spectral spaces is called spe ...
... We recall that a topological space X is spectral whenever X is sober and the compact and open subsets are closed under finite intersections and form a base for the topology of X. Note that in particular every spectral space is compact. A continuous map f : X → Y between spectral spaces is called spe ...
Article - Fundamental Research and Development
... and Nearly Closed Sets When using proper subspaces, care must be taken to ensure there are proper subspaces and that the property under consideration can be used to extend the property from a proper subset of interest to the full set X. Thus, as was true above, the property itself has a more meaning ...
... and Nearly Closed Sets When using proper subspaces, care must be taken to ensure there are proper subspaces and that the property under consideration can be used to extend the property from a proper subset of interest to the full set X. Thus, as was true above, the property itself has a more meaning ...
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.