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Problem Farm
... an interval in Cn if and only if the sequence a1 , a2 , . . . is all zeroes or all twos starting with some index (i.e. except for finitely many terms, the sequence is all zeroes or all twos) B3. Let D be the two-point set {0, 2} with the discrete topology. Show that the Cantor ...
... an interval in Cn if and only if the sequence a1 , a2 , . . . is all zeroes or all twos starting with some index (i.e. except for finitely many terms, the sequence is all zeroes or all twos) B3. Let D be the two-point set {0, 2} with the discrete topology. Show that the Cantor ...
§2.1. Topological Spaces Let X be a set. A family T of subsets of X is
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
. TOPOLOGY QUALIFYING EXAMINATION Time: Three hours.
... (a) Compute the fundamental group of the space obtained by identifying the points x and y pictured above on the solid three hole torus. (b) Compute the fundamental group of the space obtained by identifying the points x and y on the surface of the three hole torus. 2. (a) Define the join (or product ...
... (a) Compute the fundamental group of the space obtained by identifying the points x and y pictured above on the solid three hole torus. (b) Compute the fundamental group of the space obtained by identifying the points x and y on the surface of the three hole torus. 2. (a) Define the join (or product ...
COMMUTATIVE ALGEBRA HANDOUT: MORE
... define the product topology on X × Y as the topology with basis the sets U × V with U open in X and V open in Y . So a set is open in the product topology iff it is a union of such “open rectangles”. Remarks (1) If we give R and R2 the usual metrics, then the topology we get on R2 really is obtained ...
... define the product topology on X × Y as the topology with basis the sets U × V with U open in X and V open in Y . So a set is open in the product topology iff it is a union of such “open rectangles”. Remarks (1) If we give R and R2 the usual metrics, then the topology we get on R2 really is obtained ...
Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui January 12, 2008
... 4. Let X = [0, 1] × [0, 1] denote the rectangle in R2 . Let ∼ be the equivalence relation generated by (0, p) ∼ (1, 1 − p) wher 0 ≤ p ≤ 1. The quotient space X/∼ is called the Möbius band. Show that S 1 is a retract of the Möbius band. 5. Compute the first three homology groups of the hollow spher ...
... 4. Let X = [0, 1] × [0, 1] denote the rectangle in R2 . Let ∼ be the equivalence relation generated by (0, p) ∼ (1, 1 − p) wher 0 ≤ p ≤ 1. The quotient space X/∼ is called the Möbius band. Show that S 1 is a retract of the Möbius band. 5. Compute the first three homology groups of the hollow spher ...
Some point-set topology
... the empty set are closed, arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed. Given a topological space (X, T), and a subset E ⊂ X, the closure E of E is the intersection of all closed sets containing E. Said differently, E is the smallest (with respect t ...
... the empty set are closed, arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed. Given a topological space (X, T), and a subset E ⊂ X, the closure E of E is the intersection of all closed sets containing E. Said differently, E is the smallest (with respect t ...
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... Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces ...
... Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces ...
1. Prove that a continuous real-valued function on a topological
... 3. Suppose X is a topological space and A ⊆ X. Recall that A is a retract of X whenever there exists an onto continuous function X → A that is indentity on A. (a) Prove that A is a retract of X if and only if any continuous function on A can be extended to X. Let f : X → A be a retraction and suppos ...
... 3. Suppose X is a topological space and A ⊆ X. Recall that A is a retract of X whenever there exists an onto continuous function X → A that is indentity on A. (a) Prove that A is a retract of X if and only if any continuous function on A can be extended to X. Let f : X → A be a retraction and suppos ...
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... 2. If G is a topological group, G can be considered a topological transformation group on itself. There are many continuous actions that can be defined on G. For example, α : G × G → G given by α(g, x) = gx is one such action. It is continuous, and satisfies the two action axioms. G is also effectiv ...
... 2. If G is a topological group, G can be considered a topological transformation group on itself. There are many continuous actions that can be defined on G. For example, α : G × G → G given by α(g, x) = gx is one such action. It is continuous, and satisfies the two action axioms. G is also effectiv ...
Topology Ph.D. Qualifying Exam Alessandro Arsie, Gerard Thompson and Mao-Pei Tsui
... that are disjoint can be separated by disjoint open sets. Prove that in a regular space a closed set and a compact set that are disjoint can be separated by disjoint open sets. 7. Suppose A = U~A~, where each A~ is connected, and so that there is a point x common to all A~. Prove that A is connected ...
... that are disjoint can be separated by disjoint open sets. Prove that in a regular space a closed set and a compact set that are disjoint can be separated by disjoint open sets. 7. Suppose A = U~A~, where each A~ is connected, and so that there is a point x common to all A~. Prove that A is connected ...