MIDTERM 2 : Math 1700 : Spring 2014 SOLUTIONS Problem 1. (10
... function f : X → R is bounded. “⇒”: Let f : X → R be a continuous function. Since X is compact, f (X) is compact. Since compact subsets of R are bounded, f is bounded. “⇐”: Suppose that the metric space X is not compact, so that there is a sequence {xn }n∈N with no convergent subsequence. Let A be t ...
... function f : X → R is bounded. “⇒”: Let f : X → R be a continuous function. Since X is compact, f (X) is compact. Since compact subsets of R are bounded, f is bounded. “⇐”: Suppose that the metric space X is not compact, so that there is a sequence {xn }n∈N with no convergent subsequence. Let A be t ...
Exercise Sheet no. 1 of “Topology”
... For each norm k · k on Rn , the metric d(x, y) := kx − yk defines the same topology. Hint: Use that each norm is equivalent to kxk∞ := max{|xi | : i = 1, . . . , n} (cf. Analysis II). Exercise E10 Cofinite topology Let X be a set and τ := {∅} ∪ {A ⊆ X : |Ac | < ∞}. Show that τ defines a topology on ...
... For each norm k · k on Rn , the metric d(x, y) := kx − yk defines the same topology. Hint: Use that each norm is equivalent to kxk∞ := max{|xi | : i = 1, . . . , n} (cf. Analysis II). Exercise E10 Cofinite topology Let X be a set and τ := {∅} ∪ {A ⊆ X : |Ac | < ∞}. Show that τ defines a topology on ...
On Hausdorff compactifications - Mathematical Sciences Publishers
... of compactifications K(X) and K(Y) for arbitrary X and Y, and in consequence it appears that for every space X, there is a pseudocompact space Y with K(X) isomorphic to K(Y). A necessary condition for K(X) to be isomorphic to K(Y) is observed for arbitrary X and Y, and this leads to the consideratio ...
... of compactifications K(X) and K(Y) for arbitrary X and Y, and in consequence it appears that for every space X, there is a pseudocompact space Y with K(X) isomorphic to K(Y). A necessary condition for K(X) to be isomorphic to K(Y) is observed for arbitrary X and Y, and this leads to the consideratio ...
Section 17. Closed Sets and Limit Points - Faculty
... Note. The above definition is similar to the definition from Calculus 2, but the “for all ε > 0” part of Calculus 2 is replaced with “for every neighborhood” here. In the topology on X = {a, b, c} mentioned above, notice that the sequence b, b, b, . . . converges to b (because every neighborhood of ...
... Note. The above definition is similar to the definition from Calculus 2, but the “for all ε > 0” part of Calculus 2 is replaced with “for every neighborhood” here. In the topology on X = {a, b, c} mentioned above, notice that the sequence b, b, b, . . . converges to b (because every neighborhood of ...
MA4266_Lect10
... Definition Let ( X , d ), (Y , d ) be metric spaces. A function f : X Y is uniformly continuous if for every 0 there exists 0 such that d ( x, y ) d ( f ( x), f ( y )) , x, y X . Theorem 6.10: If ( X , d ) is a compact metric space and (Y , d ) is a metric space and f : X ...
... Definition Let ( X , d ), (Y , d ) be metric spaces. A function f : X Y is uniformly continuous if for every 0 there exists 0 such that d ( x, y ) d ( f ( x), f ( y )) , x, y X . Theorem 6.10: If ( X , d ) is a compact metric space and (Y , d ) is a metric space and f : X ...
Section 29. Local Compactness - Faculty
... a compact subspace of Rω that contains x ∈ Rω and there is a neighborhood of x in C, then the neighborhood contains a basis element of the form of B. But then B = [a1 , b1 ] × [a1 , a2] × · · · × [an , bn ] × R × R × · · · is a closed subspace of C and so would be compact by Theorem 26.2. But B is n ...
... a compact subspace of Rω that contains x ∈ Rω and there is a neighborhood of x in C, then the neighborhood contains a basis element of the form of B. But then B = [a1 , b1 ] × [a1 , a2] × · · · × [an , bn ] × R × R × · · · is a closed subspace of C and so would be compact by Theorem 26.2. But B is n ...
Topology Proceedings - topo.auburn.edu
... (2) Every infinite closed subset of EI is uncountable. Proof: (1) Let k : EI → I be the absolute map (see [3]) that is perfect, irreducible, θ−continuous, and onto. Let Q be a countable, dense subset of I and for each r ∈ Q, select a point dr ∈ k ← (r), and let D = {dr : r ∈ Q}. D is a countable sub ...
... (2) Every infinite closed subset of EI is uncountable. Proof: (1) Let k : EI → I be the absolute map (see [3]) that is perfect, irreducible, θ−continuous, and onto. Let Q be a countable, dense subset of I and for each r ∈ Q, select a point dr ∈ k ← (r), and let D = {dr : r ∈ Q}. D is a countable sub ...
Cantor`s Theorem and Locally Compact Spaces
... Proof. Claim: given U ⊂ X non-empty and open and x ∈ X, we can construct V ⊂ U open such that x 6∈ V . Choose a point y of U different from x (any pt of U if x 6∈ U and any point of U − {x} else–remember x is not isolated). Take disjoint nbhds W1 and W2 of x and y. Let V = W2 ∩ U . Then x 6∈ V as W1 ...
... Proof. Claim: given U ⊂ X non-empty and open and x ∈ X, we can construct V ⊂ U open such that x 6∈ V . Choose a point y of U different from x (any pt of U if x 6∈ U and any point of U − {x} else–remember x is not isolated). Take disjoint nbhds W1 and W2 of x and y. Let V = W2 ∩ U . Then x 6∈ V as W1 ...
Math 396. Gluing topologies, the Hausdorff condition, and examples
... As a fundamental example of a construction of a topology by gluing, we investigate an important example: Grassmannians. We begin with the algebraic development over a general field, and then we specialize to the case F = R where we can bring in some topology. (Everything we do over R works the same ...
... As a fundamental example of a construction of a topology by gluing, we investigate an important example: Grassmannians. We begin with the algebraic development over a general field, and then we specialize to the case F = R where we can bring in some topology. (Everything we do over R works the same ...
A remark on the extreme value theory for continued fractions
... Next we estimate the gaps between the same order level intervals. For any n ≥ N and two distinct level intervals J(τ1 , · · · , τn ) and J(σ1 , · · · , σn ) of En , we assume that J(τ1 , · · · , τn ) locates in the left of J(σ1 , · · · , σn ) without loss of generality. By Proposition 2.2, we know t ...
... Next we estimate the gaps between the same order level intervals. For any n ≥ N and two distinct level intervals J(τ1 , · · · , τn ) and J(σ1 , · · · , σn ) of En , we assume that J(τ1 , · · · , τn ) locates in the left of J(σ1 , · · · , σn ) without loss of generality. By Proposition 2.2, we know t ...
MINIMAL TOPOLOGICAL SPACES(`)
... Let (X,.T) be a minimal normal space. In order to show (X,3~) is compact, it suffices to prove that X is the same as its Cech compactification ß(X). We already know that X can be considered as a subspace of ß(X). To establish our theorem, we now offer a proof by contradiction. Assume X is not compac ...
... Let (X,.T) be a minimal normal space. In order to show (X,3~) is compact, it suffices to prove that X is the same as its Cech compactification ß(X). We already know that X can be considered as a subspace of ß(X). To establish our theorem, we now offer a proof by contradiction. Assume X is not compac ...
Answer Key
... (b) Let X be locally compact and Hausdorff, and suppose that A is a locally closed subset of X. Show that A is also locally compact and Hausdorff. SOLUTION (a) We can write [0, 1) = [0, 1] ∩ (−1, 1), so it is the intersection of an open subset and a closed subset. It is not closed in R because 1 is ...
... (b) Let X be locally compact and Hausdorff, and suppose that A is a locally closed subset of X. Show that A is also locally compact and Hausdorff. SOLUTION (a) We can write [0, 1) = [0, 1] ∩ (−1, 1), so it is the intersection of an open subset and a closed subset. It is not closed in R because 1 is ...
Metric Spaces and Topology M2PM5 - Spring 2011 Solutions Sheet
... by the definition of the subspace topology TH in H (see Exercise 2 (iii) Sheet 4), UH , VH ∈ TH , so {H, TH } is a topological Hausdorff space. (ii) Let {Ai , Ti }, i = 1, 2, be two topological Hausdorff spaces, and let T1 × T2 be endowed with the product topology (see Exercise 2 (iii) Sheet 4). Let ...
... by the definition of the subspace topology TH in H (see Exercise 2 (iii) Sheet 4), UH , VH ∈ TH , so {H, TH } is a topological Hausdorff space. (ii) Let {Ai , Ti }, i = 1, 2, be two topological Hausdorff spaces, and let T1 × T2 be endowed with the product topology (see Exercise 2 (iii) Sheet 4). Let ...
Solutions to Homework 1
... is well-defined, for if x′ ∈ X is another point with f (x′ ) = y then g(x′ ) = g(x) by hypothesis. A slightly more sophisticated proof of the converse is the following. Since f is surjective, there is a map of sets s : Y → X with f ◦ s = 1Y . (Such a map is called a “section” of f . The fact that ev ...
... is well-defined, for if x′ ∈ X is another point with f (x′ ) = y then g(x′ ) = g(x) by hypothesis. A slightly more sophisticated proof of the converse is the following. Since f is surjective, there is a map of sets s : Y → X with f ◦ s = 1Y . (Such a map is called a “section” of f . The fact that ev ...
The Hausdorff Quotient
... a quotient map qX : X → H(X) such that for each Hausdorff topological space Y and all continuous maps f : X → Y , there is a unique map f : H(X) → Y such that f = f ◦ qX . A main motivation for studying this subject is that, for as far as I know, there is no published article about the Hausdorff Quo ...
... a quotient map qX : X → H(X) such that for each Hausdorff topological space Y and all continuous maps f : X → Y , there is a unique map f : H(X) → Y such that f = f ◦ qX . A main motivation for studying this subject is that, for as far as I know, there is no published article about the Hausdorff Quo ...
Solutions for the Midterm Exam
... and so there are disjoint open sets U and V , containing x and y, respectively. By definition of the product topology, U × V is an open subset of X × X, and clearly U × V ⊂ ∆c (for otherwise U ∩ V 6= ∅). This shows that ∆c is open. Conversely, suppose ∆ is closed, that is to say, ∆c is open. Let x a ...
... and so there are disjoint open sets U and V , containing x and y, respectively. By definition of the product topology, U × V is an open subset of X × X, and clearly U × V ⊂ ∆c (for otherwise U ∩ V 6= ∅). This shows that ∆c is open. Conversely, suppose ∆ is closed, that is to say, ∆c is open. Let x a ...
Compactly generated spaces
... Proposition 2.4. If X is a WH space, then any larger topology on X is still WH. In particular, kX is still WH. Proof. Let X 0 be the set X equipped with a topology containing the original topology, i.e. the identity function id : X 0 → X is continuous. For any compact Hausdorff space K and continuou ...
... Proposition 2.4. If X is a WH space, then any larger topology on X is still WH. In particular, kX is still WH. Proof. Let X 0 be the set X equipped with a topology containing the original topology, i.e. the identity function id : X 0 → X is continuous. For any compact Hausdorff space K and continuou ...
Homework set 9 — APPM5440 — Fall 2016 From the textbook: 4.1
... that f is continuous. Prove that f is open. Prove that f does not necessarily map closed sets to closed sets. Problem 2: Prove that the co-finite topology is first countable if and only if X is countable. Problem 3: Prove that the co-finite topology on R weaker than the standard topology. The last t ...
... that f is continuous. Prove that f is open. Prove that f does not necessarily map closed sets to closed sets. Problem 2: Prove that the co-finite topology is first countable if and only if X is countable. Problem 3: Prove that the co-finite topology on R weaker than the standard topology. The last t ...
Basic Notions Of Topology
... A topological space is a pair (X, O) consisting of a set X and a topology O on X. The sets in O are called the open sets; the complement of an open subset of X is called closed. A system B of open subsets of a topological space (X, O) ist called an open basis of the topology, if every open set O ∈ O ...
... A topological space is a pair (X, O) consisting of a set X and a topology O on X. The sets in O are called the open sets; the complement of an open subset of X is called closed. A system B of open subsets of a topological space (X, O) ist called an open basis of the topology, if every open set O ∈ O ...
CONCERNING SEMI-CONTINUOUS FUNCTIONS Dragan S
... Finally, the fact that continuity implies weak-continuity gives that the Theorem A is the consequence of Corollary 2. ...
... Finally, the fact that continuity implies weak-continuity gives that the Theorem A is the consequence of Corollary 2. ...
Math 535 - General Topology Fall 2012 Homework 8 Solutions
... Solution. (Tord ≤ Tmet ) For any a ∈ R, the “open rays” (a, ∞) and (−∞, a) are metrically open. (Tmet ≤ Tord ) The metric topology on R is generated by intervals (a, b) for any a < b. But these are order-open since they are the finite intersection (a, b) = (−∞, b) ∩ (a, ∞) of “open rays”. ...
... Solution. (Tord ≤ Tmet ) For any a ∈ R, the “open rays” (a, ∞) and (−∞, a) are metrically open. (Tmet ≤ Tord ) The metric topology on R is generated by intervals (a, b) for any a < b. But these are order-open since they are the finite intersection (a, b) = (−∞, b) ∩ (a, ∞) of “open rays”. ...
COMPACT SÍ-SOUSLIN SETS ARE Ga`S result holds for f
... A set-theoretic lemma is established and used firstly to show that every countably compact ^-Souslin set is a ^ (Theorem 1). One immediate application of this result is the following. In 1945, V. Snelder [8] proved that every compact Hausdorff space with a
... A set-theoretic lemma is established and used firstly to show that every countably compact ^-Souslin set is a ^ (Theorem 1). One immediate application of this result is the following. In 1945, V. Snelder [8] proved that every compact Hausdorff space with a
The low separation axioms (T0) and (T1)
... discrete. The terminology for the hierarchy (Tn ) of separation axioms appears to have entered the literature 1935 through the influential book by Alexandroff and Hopf [3] in a section of the book called ,,Trennungsaxiome“ (pp. 58 ff.). A space is a T0 -space iff (0) for two different points there i ...
... discrete. The terminology for the hierarchy (Tn ) of separation axioms appears to have entered the literature 1935 through the influential book by Alexandroff and Hopf [3] in a section of the book called ,,Trennungsaxiome“ (pp. 58 ff.). A space is a T0 -space iff (0) for two different points there i ...
Direct limits of Hausdorff spaces
... regular Hausdorff spaces which satisfies all of the hypotheses of Theorems 1 and 2 except that the natural source maps are not dense. (They are, however, embeddings.) But the direct limit space is not Hausdorff. The following example shows that the hypotheses in the theorems that the bonding maps be ...
... regular Hausdorff spaces which satisfies all of the hypotheses of Theorems 1 and 2 except that the natural source maps are not dense. (They are, however, embeddings.) But the direct limit space is not Hausdorff. The following example shows that the hypotheses in the theorems that the bonding maps be ...
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.