Math 6210 — Fall 2012 Assignment #3 1 Compact spaces and
... Y as the set of functions from X to {0, 1}. Define a sequence in Y by fn (x) = xn . Show that this sequence has no convergent subsequence. Exercise 11. Recall from the first problem set that a topological space X is said to have a sequential topology if closed subsets of X are exactly those subsets ...
... Y as the set of functions from X to {0, 1}. Define a sequence in Y by fn (x) = xn . Show that this sequence has no convergent subsequence. Exercise 11. Recall from the first problem set that a topological space X is said to have a sequential topology if closed subsets of X are exactly those subsets ...
Topology, MM8002/SF2721, Spring 2017. Exercise set 3 Exercise 1
... Which of these statements hold for all finite products? Which for arbitrary products? Exercise 7. Consider an arbitrary product of topological spaces Πα∈A Xα . Show that the two bases: • {Πα∈A Uα | Uα is open in Xα } • {Πα∈A Uα | Uα is open in Xα and Uα 6= Xα only for finitely many α} induce non hom ...
... Which of these statements hold for all finite products? Which for arbitrary products? Exercise 7. Consider an arbitrary product of topological spaces Πα∈A Xα . Show that the two bases: • {Πα∈A Uα | Uα is open in Xα } • {Πα∈A Uα | Uα is open in Xα and Uα 6= Xα only for finitely many α} induce non hom ...
1. Prove that a continuous real-valued function on a topological
... Let f : X → A be a retraction and suppose g is a continuous function on A. Then g ◦ f is an extension of g to X. Conversely, a continuous extension of the identity on A is a retraction. (b) Prove that if X is Hausdorff, then A must be closed in X. Suppose xλ is a net in A convergent to x. Since the ...
... Let f : X → A be a retraction and suppose g is a continuous function on A. Then g ◦ f is an extension of g to X. Conversely, a continuous extension of the identity on A is a retraction. (b) Prove that if X is Hausdorff, then A must be closed in X. Suppose xλ is a net in A convergent to x. Since the ...
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... Suppose X is a completely regular topological space. Then X is said to be a P-space if every prime ideal in C(X), the ring of continuous functions on X, is maximal. For example, every space with the discrete topology is a P-space. Algebraically, a commutative reduced ring R with 1 such that every pr ...
... Suppose X is a completely regular topological space. Then X is said to be a P-space if every prime ideal in C(X), the ring of continuous functions on X, is maximal. For example, every space with the discrete topology is a P-space. Algebraically, a commutative reduced ring R with 1 such that every pr ...
Week 5: Operads and iterated loop spaces October 25, 2015
... This has a sort of converse, which is much less straightforward, and will require some substantial work: Theorem 6 (Recognition, [May72]). If Y is a connected Ck -algebra, there exists a space X and a weak equivalence of Ck -algebras Ωk Y ' X. ...
... This has a sort of converse, which is much less straightforward, and will require some substantial work: Theorem 6 (Recognition, [May72]). If Y is a connected Ck -algebra, there exists a space X and a weak equivalence of Ck -algebras Ωk Y ' X. ...
SEPARATION AXIOMS 1. The axioms The following categorization
... (3) Consider X = R with the excluded point topology T p with p = 0. This space is T0 but fails to be T1 since the only open set containing p is X. (4) Let X = R be given the lower limit topology Tll . Then X is Ti for every i = 0, 1, ..., 5. To see this is suffices to show that it is T2 and T5 (all ...
... (3) Consider X = R with the excluded point topology T p with p = 0. This space is T0 but fails to be T1 since the only open set containing p is X. (4) Let X = R be given the lower limit topology Tll . Then X is Ti for every i = 0, 1, ..., 5. To see this is suffices to show that it is T2 and T5 (all ...
The Arithmetic Square (Lecture 32)
... Definition 2. Let f : X → Y be a map of topological spaces. We say that f is a rational homotopy equivalence if it induces an isomorphism on rational cohomology H∗ (Y ; Q) → H∗ (X; Q) (this is equivalent to the assertion that f induces an isomorphism on rational homology). We say that a space Z is r ...
... Definition 2. Let f : X → Y be a map of topological spaces. We say that f is a rational homotopy equivalence if it induces an isomorphism on rational cohomology H∗ (Y ; Q) → H∗ (X; Q) (this is equivalent to the assertion that f induces an isomorphism on rational homology). We say that a space Z is r ...