Evaluation map
... Proposition 1.1. Let X be a locally compact Hausdorff space and Y an arbitrary topological space. Then the evaluation map e : X × C(X, Y ) → Y given by e(x, f ) = f (x) is continuous. Remark 1.2. The evaluation map is not always continuous. Exercise 1.3 (Munkres 46.8). Let T be a topology on the set ...
... Proposition 1.1. Let X be a locally compact Hausdorff space and Y an arbitrary topological space. Then the evaluation map e : X × C(X, Y ) → Y given by e(x, f ) = f (x) is continuous. Remark 1.2. The evaluation map is not always continuous. Exercise 1.3 (Munkres 46.8). Let T be a topology on the set ...
-closed subsets of Hausdorff spaces
... The Kate to v extension [9] (resp. Fomin extension [5]) of a Hausdorff space X is denoted as /cX(resp. σX); these ίf-closed extensions are studied in [12, 13]. In [11], it is shown that if Y is an iϊ-closed extension of X, then there is a continuous surjection / : icX—>Y such that f(x) — x for xeX. ...
... The Kate to v extension [9] (resp. Fomin extension [5]) of a Hausdorff space X is denoted as /cX(resp. σX); these ίf-closed extensions are studied in [12, 13]. In [11], it is shown that if Y is an iϊ-closed extension of X, then there is a continuous surjection / : icX—>Y such that f(x) — x for xeX. ...
Solutions to Midterm 2 Problem 1. Let X be Hausdorff and A ⊂ X
... Thus, A is also closed in X. (2) It suffices to prove that, given a closed set A ⊂ X and a function f : A → I, it admits a continuous extension g to X. Construct g by induction, using Tietze’s extension theorem. Let g1 : X1 → I be an extension of f |A∩X1 to X1 . Assuming that gn−1 : Xn−1 → I such th ...
... Thus, A is also closed in X. (2) It suffices to prove that, given a closed set A ⊂ X and a function f : A → I, it admits a continuous extension g to X. Construct g by induction, using Tietze’s extension theorem. Let g1 : X1 → I be an extension of f |A∩X1 to X1 . Assuming that gn−1 : Xn−1 → I such th ...
Hausdorff Spaces
... A regular space need not be a T1 -space. For example, the set of students in MATH 3402 together with the topology T3 generated by {S11 , S12 , S21 , S22 } is regular but not T1 . Therefore it is usual to consider regular T1 -spaces. Such spaces are denoted T3 -spaces. Every T3 -space is Hausdorff, b ...
... A regular space need not be a T1 -space. For example, the set of students in MATH 3402 together with the topology T3 generated by {S11 , S12 , S21 , S22 } is regular but not T1 . Therefore it is usual to consider regular T1 -spaces. Such spaces are denoted T3 -spaces. Every T3 -space is Hausdorff, b ...
p. 1 Math 525 Notes on section 17 Isolated points In general, a point
... In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the usual topology, sequences ...
... In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the usual topology, sequences ...
2(a) Let R be endowed with standard topology. Show that for all x ε
... Let X be a topological space and let x ε X. Suppose X is first countable, show that there exist a countable basis of x, say W = { Wn n ≥ 1 } such that Wn+1 Wn 7marks ...
... Let X be a topological space and let x ε X. Suppose X is first countable, show that there exist a countable basis of x, say W = { Wn n ≥ 1 } such that Wn+1 Wn 7marks ...
Section 31. The Separation Axioms - Faculty
... Now set K is closed because it contains its limit points (of which there are none; see Corollary 17.7): any point in R\(K ∪{0}) is in a neighborhood of the form (a, b) not intersecting K, and (−1, 1) \ K is a neighborhood of 0 which does not intersect K. We consider closed set K and point x = 0 to s ...
... Now set K is closed because it contains its limit points (of which there are none; see Corollary 17.7): any point in R\(K ∪{0}) is in a neighborhood of the form (a, b) not intersecting K, and (−1, 1) \ K is a neighborhood of 0 which does not intersect K. We consider closed set K and point x = 0 to s ...
Functional Analysis
... Problem 4 (Initial topology). Let F be a family of functions from a set X to a topological space (Y, T ). The F-initial topology Ti on X is the weakest topology such that all functions in F are continuous. a) Prove that the family of all finite intersections of sets of the form f −1 (A), where f ∈ ...
... Problem 4 (Initial topology). Let F be a family of functions from a set X to a topological space (Y, T ). The F-initial topology Ti on X is the weakest topology such that all functions in F are continuous. a) Prove that the family of all finite intersections of sets of the form f −1 (A), where f ∈ ...
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI
... Prove that the product of any non-empty class of Hausdorff Spaces is a Hausdorff Space. Prove that every compact subspace of a Hausdorff space is closed. ...
... Prove that the product of any non-empty class of Hausdorff Spaces is a Hausdorff Space. Prove that every compact subspace of a Hausdorff space is closed. ...
Partitions of unity and paracompactness - home.uni
... Remark 1.4. One could assume WLOG that the indexing set B is A. Indeed, one could add together the functions ρβ supported on the same Uα , and consider the constant zero function for each index α that does not appear as α(β). It is useful to have partitions of unity subordinate to any open cover. We ...
... Remark 1.4. One could assume WLOG that the indexing set B is A. Indeed, one could add together the functions ρβ supported on the same Uα , and consider the constant zero function for each index α that does not appear as α(β). It is useful to have partitions of unity subordinate to any open cover. We ...
A categorical characterization of CH
... This Technical Report is brought to you for free and open access by the Mellon College of Science at Research Showcase @ CMU. It has been accepted for inclusion in Department of Mathematical Sciences by an authorized administrator of Research Showcase @ CMU. For more information, please ...
... This Technical Report is brought to you for free and open access by the Mellon College of Science at Research Showcase @ CMU. It has been accepted for inclusion in Department of Mathematical Sciences by an authorized administrator of Research Showcase @ CMU. For more information, please ...
Lecture 5 Notes
... heart of analysis in Rn . If we wish to extend basic ideas of analysis to general topoligical spaces, we’d like to have some way to detect these ideas. It turns out that the so-called seperation axioms do a good job of encoding much of this information. Definition 1.1. Let X be a topological space. ...
... heart of analysis in Rn . If we wish to extend basic ideas of analysis to general topoligical spaces, we’d like to have some way to detect these ideas. It turns out that the so-called seperation axioms do a good job of encoding much of this information. Definition 1.1. Let X be a topological space. ...
1. Basic Point Set Topology Consider Rn with its usual topology and
... (B) If a surjective continuous map p : A −→ X is open or closed, prove that p is a quotient map. Give counterexamples to demonstrate that there exist quotient maps that are neither open nor closed. (C) Let q : A −→ X be a quotient map. Prove that g : X −→ Y is continuous iff g ◦ q : A −→ Y is contin ...
... (B) If a surjective continuous map p : A −→ X is open or closed, prove that p is a quotient map. Give counterexamples to demonstrate that there exist quotient maps that are neither open nor closed. (C) Let q : A −→ X be a quotient map. Prove that g : X −→ Y is continuous iff g ◦ q : A −→ Y is contin ...
A HAUSDORFF TOPOLOGY FOR THE CLOSED SUBSETS OF A
... Hausdorff topology for the space G(X) of all closed subsets of such a space X. This construction occurs naturally in the theory of C*algebras; but, in view of its purely topological nature, it seemed wise to publish it apart from the algebraic context.1 A comparison of our topology with the topology ...
... Hausdorff topology for the space G(X) of all closed subsets of such a space X. This construction occurs naturally in the theory of C*algebras; but, in view of its purely topological nature, it seemed wise to publish it apart from the algebraic context.1 A comparison of our topology with the topology ...
Convergence of Sequences and Nets in Metric and Topological
... By convergence, on an abstract level, what we attempt to determine is, as we take a walk through some kind of abstract space, is there some point to which we are heading (even if we may never reach it), or are we simply aimlessly wandering around. In order to clarify this notion, we must have first ...
... By convergence, on an abstract level, what we attempt to determine is, as we take a walk through some kind of abstract space, is there some point to which we are heading (even if we may never reach it), or are we simply aimlessly wandering around. In order to clarify this notion, we must have first ...
On Collectionwise Hausdorff Bitopological Spaces ABSTRACT 1
... In literature there are also several generalizations of the notion of Lindelöf spaces and were studied separately for different reasons and purposes. For instance, Balasubramaniam (1982) introduced and studied the notion of nearly Lindelöf spaces. Then in 1996, Cammaroto and Santoro (1966) studied a ...
... In literature there are also several generalizations of the notion of Lindelöf spaces and were studied separately for different reasons and purposes. For instance, Balasubramaniam (1982) introduced and studied the notion of nearly Lindelöf spaces. Then in 1996, Cammaroto and Santoro (1966) studied a ...
Characterizations of s-closed Hausdorff spaces
... to be semiopen [6] (respectively regular closed) if A c Cl(Int(A)) (respectively A = Cl(Int(A))). The family of all semiopen (respectively regular closed) sets in (X, x) is denoted by SO(X, x) (respectively RC(X, x)) . The complement of a semiopen set is said to be semi-closed. The intersection of a ...
... to be semiopen [6] (respectively regular closed) if A c Cl(Int(A)) (respectively A = Cl(Int(A))). The family of all semiopen (respectively regular closed) sets in (X, x) is denoted by SO(X, x) (respectively RC(X, x)) . The complement of a semiopen set is said to be semi-closed. The intersection of a ...
Midterm Exam Solutions
... If every equivalence class of ∼ is contained in either A or B, then π(A) and π(B) are disjoint and cover X/∼. Furthermore π(A) and π(B) are open, since A and B are open and each is a union of equivalence classes. Thus X/∼ = π(A) ∪ π(B) is a separation of X/∼, which is connected by hypothesis, givin ...
... If every equivalence class of ∼ is contained in either A or B, then π(A) and π(B) are disjoint and cover X/∼. Furthermore π(A) and π(B) are open, since A and B are open and each is a union of equivalence classes. Thus X/∼ = π(A) ∪ π(B) is a separation of X/∼, which is connected by hypothesis, givin ...
List 4 - Math KSU
... 4. Let (X, TX ) be a compact, metrizable space. Show that X is second-countable. 5. Let f : (X, TX ) → (Y, TY ) be a continuous map that sends open subsets to open subsets. (a) Show that if X is first-countable (respectively second-countable) then f (X) is firstcountable (respectively second-countab ...
... 4. Let (X, TX ) be a compact, metrizable space. Show that X is second-countable. 5. Let f : (X, TX ) → (Y, TY ) be a continuous map that sends open subsets to open subsets. (a) Show that if X is first-countable (respectively second-countable) then f (X) is firstcountable (respectively second-countab ...
A CLASS OF TOPOLOGICAL SPACES 1. Introduction. It is a
... is semi-regular (that is, sets of the form A~f~r form a basis for the open sets) and ü-closed (that is, closed in any Hausdorff space which contains it as a subspace). iï-closure is equivalent to the assertion that every covering by a family of open sets admits a finite subfamily whose closures form ...
... is semi-regular (that is, sets of the form A~f~r form a basis for the open sets) and ü-closed (that is, closed in any Hausdorff space which contains it as a subspace). iï-closure is equivalent to the assertion that every covering by a family of open sets admits a finite subfamily whose closures form ...
Math 535 - General Topology Additional notes
... consisting of open balls around x of rational radius. Proposition 2.4. Let X be a first-countable topological space, and A ⊆ X a subset. Let x ∈ A be in the closure of A. Then there exists a sequence {an }n∈N in A satisfying an → x. Example 2.5. The space RN with the box topology is not first-counta ...
... consisting of open balls around x of rational radius. Proposition 2.4. Let X be a first-countable topological space, and A ⊆ X a subset. Let x ∈ A be in the closure of A. Then there exists a sequence {an }n∈N in A satisfying an → x. Example 2.5. The space RN with the box topology is not first-counta ...
and x ∈ U y ∈ V ˆ N = N∪{∞} (d) Let a, b:ˆN
... U1 × X2 3 (x1 , x2 ) and U2 × X2 3 (y1 , y2 ) are disjoint open sets in X × X2 . If instead x2 6= y2 , then we use a similar argument, relying on the fact that X2 is Hausdorff. Thus X1 × X2 is Hausdorff. This argument can easily be adapted to arbitrary products: two points differ if they differ at o ...
... U1 × X2 3 (x1 , x2 ) and U2 × X2 3 (y1 , y2 ) are disjoint open sets in X × X2 . If instead x2 6= y2 , then we use a similar argument, relying on the fact that X2 is Hausdorff. Thus X1 × X2 is Hausdorff. This argument can easily be adapted to arbitrary products: two points differ if they differ at o ...
Mathematics W4051x Topology
... 1. (a) Let S and T be two topologies on the same set X with S ⊂ T . What does compactness of X under one of these topologies imply about compactness under the other? Give proofs or counterexamples. (b) Show that if X is compact Hausdorff under both S and T , then S = T . 2. (a) Show that any topolog ...
... 1. (a) Let S and T be two topologies on the same set X with S ⊂ T . What does compactness of X under one of these topologies imply about compactness under the other? Give proofs or counterexamples. (b) Show that if X is compact Hausdorff under both S and T , then S = T . 2. (a) Show that any topolog ...
p. 1 Math 490 Notes 8 Convergence, and Hausdorff and T1 Spaces
... In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the usual topology, sequences ...
... In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the usual topology, sequences ...
Solution 3
... Recall from lecture that for any topological space X with strongly discontinuous action of any group G, the map p : X → X/G is open and a covering map. Furthermore, if X is simply connected, π1 (X/G) ∼ = G. (a) To show that M/G is again a manifold, we check that it is second countable, locally Eucli ...
... Recall from lecture that for any topological space X with strongly discontinuous action of any group G, the map p : X → X/G is open and a covering map. Furthermore, if X is simply connected, π1 (X/G) ∼ = G. (a) To show that M/G is again a manifold, we check that it is second countable, locally Eucli ...
Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.