Topology HW11,5
... disjoint nonempty open subsets. Note that X is connected if only if Ø and X are the only clopen subsets of X. A subset A of X is called connected if A is a connected space with its induced topology. Clearly, a singleton subset of X is always connected. 1. Is » connected? 2. Let A be a connected subs ...
... disjoint nonempty open subsets. Note that X is connected if only if Ø and X are the only clopen subsets of X. A subset A of X is called connected if A is a connected space with its induced topology. Clearly, a singleton subset of X is always connected. 1. Is » connected? 2. Let A be a connected subs ...
Graph Topologies and Uniform Convergence in Quasi
... Similarly, the lower Hausdorff quasi-uniformity HU−−1 ×V on C(X, Y ) has basic entourages of the form (U −1 × V )− H = {(f, g) ∈ C(X, Y ) × C(X, Y ) : (U −1 × V )(x, f (x)) ∩ Gr g 6= ∅ for all x ∈ X}, where U ∈ U and V ∈ V. The Hausdorff quasi-uniformity HU −1 ×V induced on C(X, Y ) by the product q ...
... Similarly, the lower Hausdorff quasi-uniformity HU−−1 ×V on C(X, Y ) has basic entourages of the form (U −1 × V )− H = {(f, g) ∈ C(X, Y ) × C(X, Y ) : (U −1 × V )(x, f (x)) ∩ Gr g 6= ∅ for all x ∈ X}, where U ∈ U and V ∈ V. The Hausdorff quasi-uniformity HU −1 ×V induced on C(X, Y ) by the product q ...
Grade 8 Mathematics - Richland Parish School Board
... 3. Find the distance of the round trip. Record your answer in scientific notation. 456,000,000 = 4.56 x 108 km 4. The space craft will be designed to have 8,000,000 cubic feet of cargo space. Express the number in scientific notation. 8.0 x 106cubic feet 5. It has been estimated that meals for each ...
... 3. Find the distance of the round trip. Record your answer in scientific notation. 456,000,000 = 4.56 x 108 km 4. The space craft will be designed to have 8,000,000 cubic feet of cargo space. Express the number in scientific notation. 8.0 x 106cubic feet 5. It has been estimated that meals for each ...
On z-θ-Open Sets and Strongly θ-z
... spaces. Let X be a topological space and A a subset of X. The closure of A and the interior of A are denoted by cl(A) and int(A), respectively. All open neighborhoods of the point x of X is denoted by U(x). A subset A is said to be regular open (resp. regular closed) if A = int(cl(A)) (resp. A = cl( ...
... spaces. Let X be a topological space and A a subset of X. The closure of A and the interior of A are denoted by cl(A) and int(A), respectively. All open neighborhoods of the point x of X is denoted by U(x). A subset A is said to be regular open (resp. regular closed) if A = int(cl(A)) (resp. A = cl( ...
domains of perfect local homeomorphisms
... than the obvious one) will assure that the domain X is a union of finitely many components? In particular, to what degree can the requirement of [3, 1.4], that p be proper, be relaxed? Specifically, if p : X —>Y is a perfect local homeomorphism, what sorts of finiteness conditions can be obtained on ...
... than the obvious one) will assure that the domain X is a union of finitely many components? In particular, to what degree can the requirement of [3, 1.4], that p be proper, be relaxed? Specifically, if p : X —>Y is a perfect local homeomorphism, what sorts of finiteness conditions can be obtained on ...
CLOSED GRAPH THEOREMS FOR BORNOLOGICAL
... by all bounded Banach disks in E and in F , respectively, is continuous even if regarded as a mapping into the ultrabornologification Fuborn of F . The terminology and the notation of the theorem will be explained in the course of this work. It is interesting to notice that, although De Wilde’s theo ...
... by all bounded Banach disks in E and in F , respectively, is continuous even if regarded as a mapping into the ultrabornologification Fuborn of F . The terminology and the notation of the theorem will be explained in the course of this work. It is interesting to notice that, although De Wilde’s theo ...
computational topology
... A Path is a continuous function γ : [0, 1] → X . Connecting γ(0) to γ(1) in X . Self intersections are allowed (there can be s, t ∈ [0, 1] such that γ(s) = γ(t)). No self intersections ⇒ the function is injective and the path is ...
... A Path is a continuous function γ : [0, 1] → X . Connecting γ(0) to γ(1) in X . Self intersections are allowed (there can be s, t ∈ [0, 1] such that γ(s) = γ(t)). No self intersections ⇒ the function is injective and the path is ...
Chapter 2 - PSU Math Home
... coherently oriented. Vertices with only one edge are called roots if the edge is oriented away from the vertex, and leaves if it is oriented towards the vertex. 2.3.2. Planarity of graphs. The goal of this subsection is to prove that the graph K3,3 is nonplanar, i.e., possesses no topological embedd ...
... coherently oriented. Vertices with only one edge are called roots if the edge is oriented away from the vertex, and leaves if it is oriented towards the vertex. 2.3.2. Planarity of graphs. The goal of this subsection is to prove that the graph K3,3 is nonplanar, i.e., possesses no topological embedd ...
How to find a Khalimsky-continuous approximation of a real-valued function Erik Melin
... Theorem 1. A T0 smallest-neighborhood space is connected if and only if it is Khalimsky arc-connected. Proof. See for example Theorem 11 of Melin (2004). Slightly weaker is Khalimsky, Kopperman & Meyer’s (1990, Theorem 3.2c) result. On the digital plane, Z2 , the Khalimsky topology is given by the ...
... Theorem 1. A T0 smallest-neighborhood space is connected if and only if it is Khalimsky arc-connected. Proof. See for example Theorem 11 of Melin (2004). Slightly weaker is Khalimsky, Kopperman & Meyer’s (1990, Theorem 3.2c) result. On the digital plane, Z2 , the Khalimsky topology is given by the ...
Relations among continuous and various non
... not know whether k2 space and kd space are equivalent concepts. It is easy to see that a locally compact or first countable space is a k2 space. Finally the following example shows that X being k2 at a point p does not necessitate X being kx at p. EXAMPLE 4.1. This example provides a k2 space which ...
... not know whether k2 space and kd space are equivalent concepts. It is easy to see that a locally compact or first countable space is a k2 space. Finally the following example shows that X being k2 at a point p does not necessitate X being kx at p. EXAMPLE 4.1. This example provides a k2 space which ...
Some forms of the closed graph theorem
... of O(N) into G(M). As C(M) is separable, C(N) is separable and hence the closed unit ball B of O(N)' is metrizable in a((0(N))', C(N)) (Lemma 1-1). There is a natural injection of N into B continuous for cr(C(N)', C(N)) and so, as N is compact, N is homeomorphic to a subset of B; thus N is metrizabl ...
... of O(N) into G(M). As C(M) is separable, C(N) is separable and hence the closed unit ball B of O(N)' is metrizable in a((0(N))', C(N)) (Lemma 1-1). There is a natural injection of N into B continuous for cr(C(N)', C(N)) and so, as N is compact, N is homeomorphic to a subset of B; thus N is metrizabl ...
An Intuitive Introduction - University of Chicago Math Department
... one point in S n . Take any point x 6= x0 , and find a small open ball B around x. If the number of times f passes into the ball, hits x, and leaves the ball is finite, then we are done—just push each individual portion of f off of x. We must therefore show that this is in fact the case. Now, the se ...
... one point in S n . Take any point x 6= x0 , and find a small open ball B around x. If the number of times f passes into the ball, hits x, and leaves the ball is finite, then we are done—just push each individual portion of f off of x. We must therefore show that this is in fact the case. Now, the se ...
Homework Solutions 5
... 6.6l Let X = Z with the discrete topology. Then X × X = Z × Z with the discrete topology, for if m, n ∈ Z then {m} and {n} are open, so {m × n} is open in Z × Z, and every subset of Z × Z is a union of such points and thus open. Since Z and Z × Z are both countable, there is a one-to-one and onto ma ...
... 6.6l Let X = Z with the discrete topology. Then X × X = Z × Z with the discrete topology, for if m, n ∈ Z then {m} and {n} are open, so {m × n} is open in Z × Z, and every subset of Z × Z is a union of such points and thus open. Since Z and Z × Z are both countable, there is a one-to-one and onto ma ...
ON FAINTLY SEMIGENERALIZED α
... Definition 29. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is said to be θ-sgα-closed if for each (x, y) ∈ (X × Y )\G(f ), there exist U ∈ sgαO(X, x) and V ∈ σθ containing y such that (U × V ) ∩ G(f ) = ∅. Lemma 30. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is θ-sgαclosed in X × Y if and ...
... Definition 29. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is said to be θ-sgα-closed if for each (x, y) ∈ (X × Y )\G(f ), there exist U ∈ sgαO(X, x) and V ∈ σθ containing y such that (U × V ) ∩ G(f ) = ∅. Lemma 30. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is θ-sgαclosed in X × Y if and ...
Topological Spaces
... This half plane maps onto a complex plane missing the positive real axis. We identify the upper half of this cut plane with the upper half of the first cut plane, since they both represent the map of the first quadrant. However the lower half of this second cut plane is the map of the second quadran ...
... This half plane maps onto a complex plane missing the positive real axis. We identify the upper half of this cut plane with the upper half of the first cut plane, since they both represent the map of the first quadrant. However the lower half of this second cut plane is the map of the second quadran ...
On bτ-closed sets
... (4) resolvable if (X, τ ) is the union of two disjoint dense subsets. For undefined concepts we refer the reader to [10] and [9] and the references given there. 2. bτ -closed sets and their relationships In [10] the relationships between various types of generalized closed sets have been summarized ...
... (4) resolvable if (X, τ ) is the union of two disjoint dense subsets. For undefined concepts we refer the reader to [10] and [9] and the references given there. 2. bτ -closed sets and their relationships In [10] the relationships between various types of generalized closed sets have been summarized ...
ON θ-b–IRRESOLUTE FUNCTIONS 1. Introduction In 1965, Njastad
... In 1965, Njastad [7] initiated the study of so called α-open sets. This notion has been studied extensively in recent years by many topologists. As a generalization of open sets, b-open sets were introduced and studied by Andrijevic. This notions was further studied by Ekici [3, 4, 5], Park [8] and ...
... In 1965, Njastad [7] initiated the study of so called α-open sets. This notion has been studied extensively in recent years by many topologists. As a generalization of open sets, b-open sets were introduced and studied by Andrijevic. This notions was further studied by Ekici [3, 4, 5], Park [8] and ...
Italo Jose Dejter
Italo Jose Dejter (Bahia Blanca, 1939) is an Argentine-born American mathematician of Bessarabian Jewish descent. He is a professor at the University of Puerto Rico (UPRRP) since 1984 and has conducted research in mathematics, particularly in areas that include algebraic topology, differential topology, graph theory, coding theory and design theory.He has an Erdos number of 2 since 1993.Dejter completed the Licentiate degree in Mathematics from University of Buenos Aires in 1967, and the Ph.D. degree in Mathematics from Rutgers University in 1975 under the supervision of Ted Petrie. He was a professor atFederal University of Santa Catarina, Brazil, from 1977 to 1984. Dejter has been a visiting scholar at a number of research institutions, including University of São Paulo,Instituto Nacional de Matemática Pura e Aplicada, Federal University of Rio Grande do Sul,University of Cambridge,National Autonomous University of Mexico, Simon Fraser University, University of Victoria, New York University, University of Illinois at Urbana–Champaign, McMaster University,DIMACS, Autonomous University of Barcelona, Technical University of Denmark, Auburn University, Polytechnic University of Catalonia, Technical University of Madrid, Charles University, Ottawa University, Simón Bolívar University, etc. The sections below describe the relevance of Dejter's work in the research areas mentioned in the first paragraph above, or in the box to the right.