Solved MCQ on Metric Space | Open & Closed Set | Mathematical Analysis | BSc Math | Real Analysis
In this video lecture, theory and Multiple Choice Questions (MCQ) on Metric Space has been discussed in HINDI.
Must watch before any university exam or any MCQ type Exam. Very helpful and easily explained for final year students of BSc Mathematics Sant Gadge Baba Amravati University (SGBAU), Amravati and RTMNU, Nagpur. Gondwana university GU exam, SPPU Pune Online Examination solved MCQ for Mathematics. Also, helpful for Maharashtra SET Mathematics and NET Mathematical Science Examination.
BSc Semester-V
Mathematics paper X (Mathematical Analysis)
Unit 5 : Metric Space
Sgbau, amravati university winter 2021 Exam
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Timestamps:🕙
[01:00] :- Definition & Examples of Metric
[05:50] :- Neighborhood of point, limit point, interior point of set
[14:37] :- Open set and closed set
[17:07]:- MCQs on neighbourhood of point, limit point, interior point, open and closed set
[24:24] :- Union and intersection of open and closed set
[28:34] :- Cauchy Sequence and Complete metric space
[32:33] :- MCQ on Union and intersection of open and closed set and Complete metric space
In the first part of the video, the definition of metric Space along with various examples is explained. The diameter of set and distance of the point from the set is defined by using the metric. The limit point or accumulation point or cluster point is very important concept in metric Space. The neighborhood of limit point of a set contains infinitely many point of that set. A finite point set has no limit point. The interior point is a point of set such that there is neighbourhood of point that entirely lies inside the set.
open set and closed set:
A Set A is said to be open set if every point of A is an interior point of A.
A Set A is said to be closed set if every limit point of A is a point of A i.e. if it contains all its limit points.
Every neighbourhood is an open set.
A set is open if and only if its complement is closed.
Union and intersection of open and closed set :
i) Arbitrary union of open sets is open.
ii)
Finite intersection of open sets is open.
iii) Arbitrary intersection of closed sets is closed.
iv) Finite union of closed sets is closed.
Arbitrary interesction of open set need not be open and arbitrary union of closed set need not be closed.
Cauchy Sequence in Metric Space:
Every convergent sequence in a metric space is a Cauchy sequence.
But the converse of this theorem is not true. i.e. a Cauchy sequence in a metric space is not necessarily convergent.
Complete metric space:
A metric space (X,d) is said to be complete if every Cauchy sequence in X is
convergent in X.
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Metric space in Hindi, Theory and concept in metric space, score more marks in Mathematics