This document provides an overview of topics in real analysis including countable and uncountable sets, open and closed sets, connected sets, and limit points. It defines bounded and unbounded sets, with an unbounded set not being of finite size. Open sets are defined as sets containing all their limit points, while closed sets can be approximated in a metric space by containing limit points. Limit points are also defined as points that can be approximated by other points in a set's neighborhood excluding the point itself. Images are provided as examples of open and closed sets.

1. Real Analysis
15UMTC52
Mrs.P.KalaiselviM.Sc.,M.A.
Mrs.K.EswariM.Sc.,M.Phil
Department of Mathematics (SF)

2. Contents :
❏Countable and Uncountable sets
❏Open and Closed sets
❏Connected sets
❏Limit Points.

3. Bounded and Unbounded Set:
A set is called bounded, if it is,
in a certain sense, of finite size.
Conversely, a set which is not
bounded is called unbounded.

4. Bounded and Unbounded Set:

5. Definition for Open Set :

6. Images denoting open sets

7. Definition for Closed Set :
A closed set can be defined as a set which
contains all its limit points. In a complete
metric space, a closed set is a set which is
closed under the limit operation.

8. Images denoting closed sets:

12. Definition for Cluster point :
A limit point (or cluster point or accumulation
point) of a set S in a metric space X is a point x
that can be "approximated" by points of S in the
sense that every neighbourhood of x with respect
to X also contains a point of S other than x itself

14. Thank You