MATH527_HW1 - Serge Ballif MATH 527 Homework 1 September 7,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Serge Ballif MATH 527 Homework 1 September 7, 2007 Problem 1. Let X be a topological space, and A be its subset. Is it true that A 0 , the set of limit points of A is always closed in X ? It is not true in general. For a counterexample, we can take X = { a,b } with the indiscrete topology T = { X, ∅} . A point x in X is a limit point of a subset A iff every open set containing x also contains a point of A other than x itself. Consider the subset A = { a } . The point b is a limit point of A , because every open set containing b has A as a subset. However, a is not a limit point of A , since no open set containing a contains any point of A other than a . Thus A 0 = { b } , which is not closed in X . Problem 2. Construct a map f : R R that is continuous at exactly one point. Let f : R R be given by f ( x ) = ( x if x is rational - x if x is irrational We claim that f is continuous only at the point x = 0. A function f : R R is continuous at a point x iff for every sequence ( x i ) i =1 that converges to x we also have the sequence ( f ( x i )) i
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

MATH527_HW1 - Serge Ballif MATH 527 Homework 1 September 7,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online