Metric Spaces & Topology
Metric Spaces
Module 1
Metric Space
Nonempty set X with distance function d: X \times X \to \mathbb{R}, (X,d) is a metric space if d satisfies: 1. d(x,y) \geq 0 :Non-negativity 2. d(x,y) = 0 \iff x=y :Identity 3. d(x,y) = d(y,x) :Symmetricity 4. d(x,y) \leq d(x,z) + d(z,y) :Triangle Inequality \forall x,y,z \in X and x<z<y.
List of Inequalities
List of Metric Spaces
Discrete Metric Properties and Facts
- Each subset of a discrete metric space is closed.
Sequences and limits in m.s.
(X,d) be m.s. A sequence of points \{x_n\} or \{f(n)\} in X is f: \mathbb{N}\xrightarrow{into} X
Limit x: {x_n}\to x\in X if \forall \epsilon >0\exists n_0\in\mathbb{N}\nid(x_n,x)<\epsilon\quad\forall n>n_0. Remarks 1. If d is usual metric, then –continue
Cauchy sequence in m.s.
\{x_n\} is Cauchy if \forall \epsilon >0\exists n_0\in\mathbb{N}\nid(x_n,x_m)<\epsilon\quad \forall m,n>n_0.
Remarks
Complete Metric Spaces
Completion of Metric Spaces
A complete m.s. (X^*,d^*) is a completion of an incomplete m.s. (X,d) if - X is a subspace of X^*; - every point of X^* is the limit point of some sequence in X.
Module 2 Topology of Metric Spaces
Def. Open and Closed Sets/Balls
Open ball S(x_0,r)=\{x\in X: d(x_0,x)<r\} Closed ball \bar{S}(x_0,r)=\{x\in X: d(x_0,x)\leq r\} of radius r>0 and center x_0\in X.
Neighborhood
A neighborhood of the point x_0\in X is any open ball in (X, d) with centre x_0.
Open Set
G\subseteq X is open if \forall x\in G,\exists r>0\ni S(x,r)\subseteq G. i.e., each point in G is the center of some open ball contained in G. or, every point of G has a neighborhood contained in G.
Interior Point of a Set
A\subseteq X, x\in X is interior point of A if \exists S(x,r)\subseteq A for some r>0. or, x\in X has a neighborhood contained in A.
Interior Set
Set of all interior points of A is called interior of A Int(A)=A^{\circ}=\{x\in A: S(x,r)\subseteq A for some r>0\}.
Properties of Interior Set
- Int(A)\subseteq A
Limit Point of a Set
F\subseteq X, x\in X is a limit point of F if each neighborhood of x contains at least one point of F different from x. (S(x,r)-\{x\})\cap F\neq\phi
Derived Set
Set of all limit points F' is called derived set of F. F'=\{x\in X: (S(x,r)-\{x\})\cap F\neq\phi\}
Closed Set
F\subseteq X is closed if it contains each of its limit points i.e., F'\subseteq F.
Closure of a Set
F\subseteq X, \bar{F}=F\cup F' is called the closure of F.
Properties of Closure
- \bar{F} is closed
Bounded Set
Nonempty A\subseteq X, A is bounded if \exists M>0\nid(x,y)\leq M,\quad x,y\in A
![[Pasted image 20251111194405. ### Smallest Closed Interval Nonempty F\in B(\mathbb{R}), let \alpha =\inf F and \beta = \sup F. [\alpha, \beta] is called the smallest closed interval containing F.
Cantor
Diameter of a Set
If A\in B, diam(A)=d(A)=\sup\{d(x,y): x,y\in A\}. If A\notin B, d(A)=\infty
Distance between point and a subset
Between x\in X and B\subseteq X, d(x,B)=\inf\{d(x,y):y\in B\}. #### Distance between two subsets Between B\subseteq X and C\subseteq X, d(B,C)=\inf\{d(x,y):x\in B,y\in C\}.
Cantor’s Theorem
Subspaces
Nonempty Y\subseteq X. d_Y be the restriction of d to Y\times Y, then (Y,d_Y) is a m.s. d_Y is the metric induced by d on subspace Y.
Dense Set
X_0\subseteq X is everywhere dense or simply dense if \bar{X_0}=X. i.e., if every point of X is either a point or a limit point of X_0. or \forall x\in X, \exists \{x_{0_n}\} \to x.
- X_0 is dense \iff (X_0^c)^\circ =\phi.
- Trivially, X is always a dense subset of itself.
Module 3
Continuity in Metric Spaces
(X,d_X), (Y,d_Y) be m.s., A\subseteq X. f:A\to Y, f\in C(a\in A) if \forall \epsilon>0,\exists \delta>0\nid_Y(f(x),f(a))<\epsilon,\quad\forall x\in A\land d_X(x,a)<\delta. If f\in C(a\in A)\forall a\in A\implies f\in C(A).