# Metric Spaces (Notes)

These are updated version of previous notes. Many mistakes and errors have been removed. These notes are collected, composed and corrected by [[::Atiq]]. 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, PhD |

Lectures | Prof. Muhammad Ashfaq |

Pages | 24 pages |

Format | |

Size | 275KB |

### CONTENTS OR SUMMARY:

- Metric Spaces and examples
- Pseudometric and example
- Distance between sets
- Theorem: Let $(X,d)$ be a metric space. Then for any $x,y\in X$, $$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$
- 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 $A^c$ is open. - Theorem: A closed ball is a closed set.
- Theorem: Let (
*X,d*) be a metric space and $A\subset X$. If $x \in X$ is a limit point of*A*. Then every open ball $B(x;r)$ 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, $A \subset X$ 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: $f:(X,d)\to (Y,d')$ is continuous at $x_0\in X$ if and only if $f^{-1}(G)$ is open is
*X*. wherever*G*is open in*Y*. - Convergence of Sequence
- Theorem: If $(x_n)$ is converges then limit of $(x_n)$ is unique.
- Theorem: (i) A convergent sequence is bounded. (ii) ii) If ${x_n}\to x$ and ${y_n}\to y$ then $d(x_n,y_n)\to d(x,y)$.
- Cauchy Sequence
- Theorem: A convergent sequence in a metric space (
*X,d*) is Cauchy. - Subsequence
- Theorem: (i) Let $(x_n)$ be a Cauchy sequence in (
*X,d*), then $(x_n)$ converges to a point $x\in X$ if and only if $(x_n)$ has a convergent subsequence $\left(x_{n_k}\right)$ which converges to $x\in X$. - (ii) If $(x_n)$ converges to $x\in X$, then every subsequence $\left(x_{n_k}\right)$ also converges to $x\in X$.
- Theorem: Let (
*X,d*) be a metric space and $M \subseteq X$. (i) Then $x\in\overline{M}$ if and only if there is a sequence $(x_n)$ in*M*such that $x_n\to x$. (ii) If for any sequence $(x_n)$ in*M*, ${x_n}\to x\quad\Rightarrow\quad x\in M$, 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 $\mathbb{R}^n$ is complete.
- Theorem: The space $l^{\infty}$ is complete.
- Theorem: The space
**C**of all convergent sequence of complex number is complete. - Theorem: The space $l^p,p\ge1$ is a real number, is complete.
- Theorem: The space
**C**[a, b] is complete. - Theorem: If $(X,d_1)$ and $\left(Y,d_2\right)$ are complete then $X\times Y$ is complete.
- Theorem: $f:\left(X,d\right)\to\left(Y,d'\right)$ is continuous at $x_0\in X$ if and only if $x_n\to x$ implies $f(x_n)\to f(x_0)$.
- Rare (or nowhere dense in
*X*) - Meager (or of the first category)
- Non-meager (or of the second category)
- Bair’s Category Theorem: If $X\ne\phi$ is complete then it is non-meager in itself “OR” A complete metric space is of second category.

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