Infinite subset of compact set has a limit point in set | Compactness | Theorem | Real analysis

Infinite subset of compact set has a limit point in set | Compactness in Real Analysis | Compact Set | Real analysis | Compactness theorem | metric space | Basic Topology | Math tutorials | Classes By Cheena Banga

***Open Cover and sub cover of a set | Definition with Examples.***
   • Open cover and Sub cover | Finite Sub...  
***Real Analysis playlist***
   • Real Analysis  
***Compactness and connectedness in real analysis definition and theorems | playlist***
   • Compactness | Connectedness | Theorem...  

useful for Msc | BSC | NET | NBHM | LPU | DU | IIT JAM | TIFR

Other topics covered in playlist:
Heine-Borel theorem
Closed Set | definition | theorems
set is closed iff its complement is open
Bolzano weierstrass theorem : Every infinite bounded subset of R has a limit point.
Definition of Neighbourhood of a point
Definition of Open set
infinite intersection of open sets need not to be open
Union of two NBDS is NBD
Intersection of NBDS is NBD
Superset of a NBD is also a NBD
Every Open interval (a,b) is neighbourhood of each of its points.
Closed interval is neighbourhood of each point except end points.
real numbers is NBD of each real number
Rational numbers set is not the neighbourhood of any of its points.
Metric space | Distance Function | Example
Metric space : Definition and Axioms
Real Analysis : Introduction and Intervals
Union of countable sets is countable
Finite,infinite,equivalent,denumerable,countable sets
Infinite subset of countable set is countable
Field,Ordered Field,complete Ordered Field
Set of Integers is Countable
Supremum and infimum
Set is countably infinite iff it can be written in the form distinct elements
Continuum Hypothesis
Cartesian product of two countable sets is Countable
Set of Rational numbers is Countable


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Math Tutorials
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Definition of metric Space
Examples of metric space
Open and Closed sets
Topology and convergence
Types of metric spaces
Complete Spaces
Bounded and complete bounded spaces
Compact spaces
Locally compact and proper spaces
connectedness
Separable spaces
Pointed Metric spaces
Types of maps between metric spaces
continuous maps
uniformly continuous maps
Lipschitz-continuous maps and contractions
isometries
Quasi-isometries
notions of metric space equivalence
Topological properties
Distance between points and sets
Hausdorff distance and Gromov metric
Product metric spaces
Continuity of distance
Quotient metric spaces
Generalizations of metric spaces
Metric spaces as enriched categories

Compactness in Real analysis
compactness in metric space
compactness in topology
compactness and connectedness in real analysis
compactness and connectedness
compactness in topological space
Connectedness in Real analysis
connectedness in metric space
connectedness in topology
connectedness in topological space
Theorems on connectedness
theorems on compactness