Metric Spaces (Notes)

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.

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: $f:(X,d)\to (Y,d^{\prime })$ is continuous at $x_0\in X$ if and only if $f^{-1}\left(G\right)$ is open is X. wherever G is open in Y.
• Convergence of Sequence
• Theorem: If $\left(x_n\right)$ is converges then limit of $\left(x_n\right)$ 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 $\left(x_n\right)$ be a Cauchy sequence in (X,d), then $\left(x_n\right)$ converges to a point $x\in X$ if and only if $\left(x_n\right)$ has a convergent subsequence $\left(x_{n_k}\right)$ which converges to $x\in X$.
• (ii) If $\left(x_n\right)$ 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 $\left(x_n\right)$ in M such that $x_n\to x$. (ii) If for any sequence $\left(x_n\right)$ in M, $x_n\to x\Rightarrow x\in M$, 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 $R^n$ 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 $l^p,p\ge 1$ 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: $f:\left(X,d\right)\to \left(Y,d^{\prime }\right)$ is continuous at $x_0\in X$ if and only if $x_n\to x$ implies $f\left(x_n\right)\to f\left(x_0\right)$.
• Rare (or nowhere dense in X)
• Meager (or of the first category)
• Non-meager (or of the second category)
• Bair’s Category Theorem: If $X\ne \varphi$ is complete then it is non-meager in itself “OR” A complete metric space is of second category.