====== Metric Spaces (Notes) ======
{{ :notes:metric-space-notes.jpg?nolink&600|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]]  |
^ 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 $(X,d)$ 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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====There are other notes on the Metric Spaces====
{{topic>Metric_Space&nouser&simplelist}}

{{tag>BS MSc Notes Metric_Space}}