THE UNIVERSITY OF TOLEDO Topology M.A. Comprehensive Examination April , 2010
... This exam has been checked carefully for errors. I.f you find what you believe to be an er’ro’r in a question, report this to the proctor. If the proctor’s inte~pretation still seems unsatisfactory to you, you may alter the question so that in your view it is correctly stated, but not in such a way ...
... This exam has been checked carefully for errors. I.f you find what you believe to be an er’ro’r in a question, report this to the proctor. If the proctor’s inte~pretation still seems unsatisfactory to you, you may alter the question so that in your view it is correctly stated, but not in such a way ...
2 A topological interlude
... in Y , there are disjoint open sets G1 , G2 ⊆ Y with y1 ∈ G1 and y2 ∈ G2 . It is not hard to show that any compact subset of a Hausdorff space is closed. If X and Y are topological spaces then a map θ : X → Y is continuous if, for all open sets G ⊆ Y , the set θ−1 (G) is open in X. Taking complement ...
... in Y , there are disjoint open sets G1 , G2 ⊆ Y with y1 ∈ G1 and y2 ∈ G2 . It is not hard to show that any compact subset of a Hausdorff space is closed. If X and Y are topological spaces then a map θ : X → Y is continuous if, for all open sets G ⊆ Y , the set θ−1 (G) is open in X. Taking complement ...
Metric Topology, ctd.
... we have x ∈ f −1 (BdY (f (x), )). As the preimage is open in X, we can find δ > 0 such that BdX (x, δ) ⊂ f −1 (BdY (f (x), )). Then dX (x, x0 ) < δ implies that dY (f (x), f (x0 )) < . Suppose that the -δ condition holds. Let V ⊂ Y be open. We show that the preimage f −1 (V ) is open. If f −1 (V ...
... we have x ∈ f −1 (BdY (f (x), )). As the preimage is open in X, we can find δ > 0 such that BdX (x, δ) ⊂ f −1 (BdY (f (x), )). Then dX (x, x0 ) < δ implies that dY (f (x), f (x0 )) < . Suppose that the -δ condition holds. Let V ⊂ Y be open. We show that the preimage f −1 (V ) is open. If f −1 (V ...
Homework 5 (pdf)
... (1) Let X be a topological space. If C is a finite subset of X, show that C is compact. (2) Let X be a set with the discrete topology. If C ⊆ X is compact, show that C is finite. (3) Let X be a topological sapce. Show that X is a T1 -space if and only if each point of X is a closed set. (4) Give a d ...
... (1) Let X be a topological space. If C is a finite subset of X, show that C is compact. (2) Let X be a set with the discrete topology. If C ⊆ X is compact, show that C is finite. (3) Let X be a topological sapce. Show that X is a T1 -space if and only if each point of X is a closed set. (4) Give a d ...
p. 1 Math 490 Notes 7 Zero Dimensional Spaces For (SΩ,τo
... B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. The sets in B are τo -open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. Sets which are simultaneously open and closed relative ...
... B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. The sets in B are τo -open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. Sets which are simultaneously open and closed relative ...
Review of basic topology concepts
... A map cl with the above properties is called a closure operator. Definition. Suppose (X, T ) is a topological space. Assume A ⊂ X is a subset of X. On A we can introduce a natural topology, sometimes denoted by T |A which consists of all subsets of A of the form A ∩ U with U open set in X. This topo ...
... A map cl with the above properties is called a closure operator. Definition. Suppose (X, T ) is a topological space. Assume A ⊂ X is a subset of X. On A we can introduce a natural topology, sometimes denoted by T |A which consists of all subsets of A of the form A ∩ U with U open set in X. This topo ...
Lecture 2
... • R with the euclidean topology is separable. • The space C([0, 1]) of all continuous functions from [0, 1] to R endowed with the uniform topology is separable, since by the Weirstrass approximation theorem Q[x] = C([0, 1]). Let us briefly consider now the notion of convergence. First of all let us ...
... • R with the euclidean topology is separable. • The space C([0, 1]) of all continuous functions from [0, 1] to R endowed with the uniform topology is separable, since by the Weirstrass approximation theorem Q[x] = C([0, 1]). Let us briefly consider now the notion of convergence. First of all let us ...
Complete three of the following five problems. In the next... assumed to be a topological space. All “maps” given in...
... assumed to be a topological space. All “maps” given in both sections are assumed to be continuous although in a particular problem you may need to establish continuity of a particular map. 1. A subset A ⊂ X is said to be locally closed if for any x ∈ X there is a neighborhood U of x such that A ∩ U ...
... assumed to be a topological space. All “maps” given in both sections are assumed to be continuous although in a particular problem you may need to establish continuity of a particular map. 1. A subset A ⊂ X is said to be locally closed if for any x ∈ X there is a neighborhood U of x such that A ∩ U ...
RECOLLECTIONS FROM POINT SET TOPOLOGY FOR
... (1) If (X, T ) is a Hausdorff space and a sequence {pn } converges to p and q then p = q. (2) If (X, T ) is a first countable space then a point p is a limit point of a set A if and only if there is a sequence {pn } in A such that pn → p. (3) Metric topological spaces are Hausdorff and first countab ...
... (1) If (X, T ) is a Hausdorff space and a sequence {pn } converges to p and q then p = q. (2) If (X, T ) is a first countable space then a point p is a limit point of a set A if and only if there is a sequence {pn } in A such that pn → p. (3) Metric topological spaces are Hausdorff and first countab ...
June 2012
... 1) Assume that (X, τ ) is a topological space with the property that for every open set G ⊆ X, the closure of G, G, is open. Such topological spaces are called extremally disconnected. Prove the following. a) If F ⊆ X is a closed set, then the interior of F , F ◦ , is closed. b) If G ⊆ X is an open ...
... 1) Assume that (X, τ ) is a topological space with the property that for every open set G ⊆ X, the closure of G, G, is open. Such topological spaces are called extremally disconnected. Prove the following. a) If F ⊆ X is a closed set, then the interior of F , F ◦ , is closed. b) If G ⊆ X is an open ...
PDF
... mathematics are directly related to the core concept of discrete space: • A discrete set is a set that, endowed with the topology implied by the context, is a discrete space. For instance for a subset of Rn and without information suggesting otherwise, the topology on the set would be assumed the us ...
... mathematics are directly related to the core concept of discrete space: • A discrete set is a set that, endowed with the topology implied by the context, is a discrete space. For instance for a subset of Rn and without information suggesting otherwise, the topology on the set would be assumed the us ...
Math 8301, Manifolds and Topology Homework 7
... 1. (CORRECTED) Let F = hx, yi be a free group on two generators. Show explicitly that the subgroup H ⊂ F generated by the elements zn = (y −n xy n ), as n ranges over the integers, is a free group on the elements zn . 2. If F and F 0 are free groups, show that F ∗ F 0 is also free. 3. Suppose f : H ...
... 1. (CORRECTED) Let F = hx, yi be a free group on two generators. Show explicitly that the subgroup H ⊂ F generated by the elements zn = (y −n xy n ), as n ranges over the integers, is a free group on the elements zn . 2. If F and F 0 are free groups, show that F ∗ F 0 is also free. 3. Suppose f : H ...
Operator Compactification of Topological Spaces
... Now we give the definition of β-compactification of a topological space. Definition 5. Let (X, Γ) and (Y, Ψ) be two topological spaces, α and β operators associated with Γ and Ψ, respectively. We say that (Y, Ψ) is a β-compactification of X if 1. (Y , Ψ) is β-compact. 2. (X, Γ) is a subspace of (Y, ...
... Now we give the definition of β-compactification of a topological space. Definition 5. Let (X, Γ) and (Y, Ψ) be two topological spaces, α and β operators associated with Γ and Ψ, respectively. We say that (Y, Ψ) is a β-compactification of X if 1. (Y , Ψ) is β-compact. 2. (X, Γ) is a subspace of (Y, ...