Topology
... Master’s Exam January 10, 2007 Instructions: Work at most one problem per side of the furnished paper. (1) Suppose A, X, and Y are topological spaces. Give X × Y the product topology. Suppose πX πY πX : X × Y → X and πY : X × Y → Y are given by (x, y) 7→ x and (x, y) 7→ y. Prove that a function f : ...
... Master’s Exam January 10, 2007 Instructions: Work at most one problem per side of the furnished paper. (1) Suppose A, X, and Y are topological spaces. Give X × Y the product topology. Suppose πX πY πX : X × Y → X and πY : X × Y → Y are given by (x, y) 7→ x and (x, y) 7→ y. Prove that a function f : ...
Homework 5 (pdf)
... (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 direct proof that a metric space (X, d) is Hausdorff. (Do not for example use the fact that a metric space is a T3 -space and every T3 -space is a T2 -space.) (5) Let f : X → Y be ...
... (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 direct proof that a metric space (X, d) is Hausdorff. (Do not for example use the fact that a metric space is a T3 -space and every T3 -space is a T2 -space.) (5) Let f : X → Y be ...
PDF
... that f −1 is also continuous. We also say that two spaces are homeomorphic if such a map exists. If two topological spaces are homeomorphic, they are topologically equivalent — using the techniques of topology, there is no way of distinguishing one space from the other. An autohomeomorphism (also kn ...
... that f −1 is also continuous. We also say that two spaces are homeomorphic if such a map exists. If two topological spaces are homeomorphic, they are topologically equivalent — using the techniques of topology, there is no way of distinguishing one space from the other. An autohomeomorphism (also kn ...
Homework 4
... II. (a) Show that (−∞, a) ∪ [a, +∞) is a separation of the space R` for any real a. (b) Describe connected components of R` and classify all continuous maps R −→ R` . Munkres exercise 3 on page 152. Exercises on pages 157-158: • Exercise 1 (imbeddings are defined on page 105) • Exercise 2 (Hint: fir ...
... II. (a) Show that (−∞, a) ∪ [a, +∞) is a separation of the space R` for any real a. (b) Describe connected components of R` and classify all continuous maps R −→ R` . Munkres exercise 3 on page 152. Exercises on pages 157-158: • Exercise 1 (imbeddings are defined on page 105) • Exercise 2 (Hint: fir ...
Products and quotients via universal property
... This exercise consists of reinterpreting some of the results we’ve proved in class. You may use results in the book without having to reprove them. 1. Let X and Y be topological spaces. Prove that X × Y has the following universal property: if Z is a topological space and fX : Z → X and fY : Z → Y a ...
... This exercise consists of reinterpreting some of the results we’ve proved in class. You may use results in the book without having to reprove them. 1. Let X and Y be topological spaces. Prove that X × Y has the following universal property: if Z is a topological space and fX : Z → X and fY : Z → Y a ...
