# Exercise 0.13. Let X be a metric space and (xn) C X a sequence. Sho...

## Transcribed Text

Exercise 0.13. Let X be a metric space and (xn) C X a sequence. Show that (xn) converges to the point x € X if and only if the sequence of real numbers d(xn,x) converges to zero.Exercise 0.15. Given topological spaces X and Y, show that the (first) pro- jection map TT : X x Y X defined by (x,y) I is continuous Exercise 0.17. Given topological spaces X, Y and Z, suppose the functions f: X Y and g: X Z are continuous. Prove the following function is continuous: h: X Y X Z I (f(x),g(x)) when the codomain carries the product topology. Exercise 0.18. Let (X,dx) and (Y,dy) be metric spaces. We can define a function (xxY)² R as follows: = This is called the product metric. Prove it is indeed a metric, and that it generates the product topology. Thus show every product of metric spaces is a metric space.

## Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

By purchasing this solution you'll be able to access the following files:
HmJan26.pdf and Solution.pdf.

\$13.00
for this solution

or FREE if you
register a new account!

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Topology Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.