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| - | ====== Metric Spaces (Notes) ====== | ||
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| - | 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, | ||
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| - | 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. | ||
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| - | ^ Name |Metric Spaces (Notes) - Version 2 | | ||
| - | ^ Author | ||
| - | ^ Lectures | ||
| - | ^ Pages |24 pages | | ||
| - | ^ Format | ||
| - | ^ Size |275KB | ||
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| - | ==== 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| {\, | ||
| - | * 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, | ||
| - | * Convergence of Sequence | ||
| - | * Theorem: If $(x_n)$ | ||
| - | * Theorem: (i) A convergent sequence is bounded. (ii) ii) If ${x_n}\to x$ and ${y_n}\to y$ then $d(x_n, | ||
| - | * 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, | ||
| - | * Theorem: $f: | ||
| - | * 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 " | ||
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| - | ==== Download or View online ==== | ||
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| - | * **[[pdf> | ||
| - | * **To view online at Scribd %%[%%[[http:// | ||
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