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 ...

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 ...

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... 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 ...