#Every compact subset of metric space is closed and bounded#UPSC mathematics optional#B.Sc.3rdyr#L40

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topic cover

every compact subset of a metric space is closed and bonded.

topic cover in lecture 39
Every compact metric space has bolzano weierstrass property

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sequentially compact
Bolzano Weierstrass Property
A metric space is sequentially compact iff it has bolzano weierstrass prperty.

topic cover in lecture 37
union of two compact subset of metric space is compact.

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compact space
every closed subset of compact metric space is compact .

topic cover in lecture 35
f is continuous iff image of closure of A is subset of closure of image of A, A is subset of X

topic cover in 34
inverse image of open is open iff function is continuous

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inverse image of open is open iff function is continuous

topic cover in lecture 32
Density Theorem
between every two different real no. there exist atleast one rational no.

topic cover in lecture 31
Archimedean Property in real, with proof

topic cover in lecture 30
property of supremum
sup (S+T) = sup S + sup T

topic cover in lecture 29
properties of mode
|x+y| less than equal to |x|+|y|
||x|-|y|| less than equal to |x-y|

topic cover in lecture 28
real no. as complete ordered field
√2 is not a rational no.

topic cover in lecture 27
completeness in R
every (R,d) metric space is complete where d is usual metric space

topic cover in lecture 26
complete metric space
if (X,d) is complete metric space and Y is a subspace of X then if Y is complete iff Y is closed.

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every cauchy sequence in a metric space is bounded.
E is a subset of metric space (X,d). E is bounded iff diameter of E is finite real number.

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cauchy sequence
convergent sequence
Every convergent sequence in a metric space is cauchy. but its converse is not true.

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definition : closure set
interior of set
bounded set
Every convergent sequence in a metric space is bounded.

Topic cover in lecture 22
in a metric space, union of finite number of closed set is closed.
in a metric space, a set is closed if and only if it contains its set of limit point.

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in metric space, every closed sphere is a closed set.
in metric space, the intersection of an arbitrary collection of closed set is closed.

topic cover in lecture 20
In metric space, the intersection of a finite number of open sets is open.

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In a metric space, every open sphere is an open set.

topic cover in lecture 18
some more examples : to solve metric

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Very well explained ☺️ thank u so much mam 🙏

KirtiRawat_

Mam last line m ...2in upper bound of d (x, y) nh ho ga?

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