Metric Spaces (Notes)

triangle inequality in a metric space These are updated version of previous notes. Many mistakes and errors have been removed. These notes are collected, composed and corrected by Atiq ur Rehman. These are actually based on the lectures delivered by Prof. Muhammad Ashfaq (Ex HoD, Department of Mathematics, Government College Sargodha).

These notes are very helpful to prepare a section of paper mostly called Topology in MSc for University of the Punjab and University of Sargodha. These are also helpful in BSc.

Name	Metric Spaces (Notes) - Version 2
Author	Atiq ur Rehman
Lectures	Prof. Muhammad Ashfaq
Pages	24 pages
Format	PDF
Size	275KB

CONTENTS OR SUMMARY:

Metric Spaces and examples
Pseudometric and example
Distance between sets
Theorem: Let $\small (X,d)$ be a metric space. Then for any $\small x,y\in X$ , $\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$
Diameter of a set
Bounded Set
Theorem: The union of two bounded set is bounded.
Open Ball, closed ball, sphere and examples
Open Set
Theorem: An open ball in metric space X is open.
Limit point of a set
Closed Set
Theorem: A subset A of a metric space is closed if and only if its complement $\small A^c$ is open.
Theorem: A closed ball is a closed set.
Theorem: Let (X,d) be a metric space and $\small A\subset X$ . If $\small x \in X$ is a limit point of A. Then every open ball $\small B(x;r)$ with centre x contain an infinite numbers of point of A.
Closure of a Set
Dense Set
Countable Set
Separable Space
Theorem: Let (X,d) be a metric space, $\small A \subset X$ is dense if and only if A has non-empty intersection with any open subset of X.
Neighbourhood of a Point
Interior Point
Continuity
Theorem: $\small f:(X,d)\to (Y,d')$ is continuous at $\small x_0\in X$ if and only if $\small f^{-1}(G)$ is open is X. wherever G is open in Y.
Convergence of Sequence
Theorem: If $\small (x_n)$ is converges then limit of $\small (x_n)$ is unique.
Theorem: (i) A convergent sequence is bounded. (ii) ii) If $\small {x_n}\to x$ and $\small {y_n}\to y$ then $\small d(x_n,y_n)\to d(x,y)$ .
Cauchy Sequence
Theorem: A convergent sequence in a metric space (X,d) is Cauchy.
Subsequence
Theorem: (i) Let $\small (x_n)$ be a Cauchy sequence in (X,d), then $\small (x_n)$ converges to a point $\small x\in X$ if and only if $\small (x_n)$ has a convergent subsequence $\small \left(x_{n_k}\right)$ which converges to $\small x\in X$ .
(ii) If $\small (x_n)$ converges to $\small x\in X$ , then every subsequence $\small \left(x_{n_k}\right)$ also converges to $\small x\in X$ .
Theorem: Let (X,d) be a metric space and $\small M subset X$ . (i) Then $\small x\in\overline{M}$ if and only if there is a sequence $\small (x_n)$ in M such that $\small x_n\to x$ . (ii) If for any sequence $\small (x_n)$ in M, $\small {x_n}\to x\quad\Rightarrow\quadx\in M$ , then M is closed.
Complete Space
Subspace
Theorem: A subspace of a complete metric space (X,d) is complete if and only if Y is closed in X.
Nested Sequence
Theorem (Cantor’s Intersection Theorem): A metric space (X,d) is complete if and only if every nested sequence of non-empty closed subset of X, whose diameter tends to zero, has a non-empty intersection.
Complete Space (Examples)
Theorem: The real line is complete.
Theorem: The Euclidean space $\small \mathbb{R}^n$ is complete.
Theorem: The space $\small l^{\infty}$ is complete.
Theorem: The space C of all convergent sequence of complex number is complete.
Theorem: The space $\small l^p,p\ge1$ is a real number, is complete.
Theorem: The space C[a, b] is complete.
Theorem: If $\small (X,d_1)$ and $\small \left(Y,d_2\right)$ are complete then $\small X\times Y$ is complete.
Theorem: $\small f:\left(X,d\right)\to\left(Y,d'\right)$ is continuous at $\small x_0\in X$ if and only if $\small x_n\to x$ implies $\small f(x_n)\to f(x_0)$ .
Rare (or nowhere dense in X)
Meager (or of the first category)
Non-meager (or of the second category)
Bair’s Category Theorem: If $\small X\ne\phi$ is complete then it is non-meager in itself “OR” A complete metric space is of second category.

Download or View online

Download PDF (275KB) | View Online
Download from 4shared.com
To view online at Scribd [Click Here]