This is short article which gives explores the beauty and importance of open sets

1. Exploring the importance of open sets
History Trivia
"The name open set was originated by Lebesgue in his doctoral dissertation of 1902, whose
essential aim was to introduce Lebesgue measure (as an extension of the Borel-measurable sets)
and the Lebesgue integral."
If we try to abstract the idea of a metric space to one without a distance function, then we
need to come up with something in a metric space that doesn't depend on precise distances.
In particular, open sets capture this idea
Intuitively speaking, an open set is a set without a border , every element of the set has,
in its neighborhood, other elements of the set. If, starting from a point of the open set, you
move away a little, you never exit the set. Roughly speaking , open set is a set where you are
able not to leave it if you move just a little bit.
In a more formal way a set X is open if for every point p in X, there exists a neighborhood
(open ball) N of p such that N is a subset of X. Concept or idea behind open set, in
general, is very abstract: any collection of sets can be called open, as long as the union of
an arbitrary number of open sets is open, the intersection of a

2. nite number of open sets is
open, and the space itself is open.
The rigorous dei

3. nitions of open sets is fundamental to topology ,We de

4. ne topology by saying
that what the open sets are topology is determined by what we call open sets and not by the
space. For example, another topology on the real line de

5. nes a set to be open if its complement
has only a

6. nite number of points(co-complement topolgy).!!!
Open sets are typically used as domains for functions, as they are more useful for analysing
"continuous" properties like dierentiability, otherwise at end points slope will not be uniquely
de

7. ned .For the notion of dierentiabilty at some point , we strongly require aur function to be
de

8. ned in some neighbourhood of x , So you need x in the interior of the domain of de

9. nition
of f otherwise limit will not make sense , To make sense of the limit, we explicitly require that
f be de

10. ned on an open interval containing x. for example consider the function de

11. ned on
singelton set .
Also they don't have borders (hence we don't have to deal or worry whether function will
be de

12. ned on other side of edge or not . )
Open set provides a method to distinguish two points. For example, if about one point in
a topological space there exists an open set not containing another (distinct) point, the two
points are referred to as topologically distinguishable.(wikipedia)
Open sets are related to the notion of closeness and convergence ,We can study the conver-
gence and continuity of function in terms of open sets . The notion of an open set provides a
fundamental way to speak of nearness of points in a topological space, without explicitly having
a concept of distance de

13. ned
If there is one theorem in analysis that is now inseparably linked to open sets, it is the
1

14. result that in English-speaking countries and in Germany is called the HeineBorel Theorem
but that in France is called the BorelLebesgue Theorem. The latter name is more appropriate,
but the history of this theorem is quite complicated and so we defer it to another occasion.
The theorem uses a concept originally called bicompactness when

15. rst formulated during the
1920s but now universally known as compactness: A set E is said to be compact if, given any
family S of open sets that covers E, some

16. nite subset of S covers E.
The theorem then states that every closed and bounded set of real numbers is compact. The
concept of an open set is essential to stating compactness, and hence is essential to the theorem
in question in full generality.(http://www.sciencedirect.com/science/article/pii/S0315086008000050)
A word of caution !!
Open is relative
Whether a set is open or not depend on the topology under consideration.Consider the
following example. Let G is de

17. ned as the set of rational numbers in (0; 1) then G is an
open subset of rational numbers but not of real numbers.
Open and closed are not mutually exclusive
{ We can have a set which is both open and closed !!! In any topology, the entire set
X is declared open by de

18. nition, as is the empty set. Moreover, the complement of
the entire set X is the empty set. So it is both open and closed.
{ Consider , since neither K = [0; 1) nor its complement KC = (1; 0) [ [1;1)
belongs to the Euclidean topology (neither one can be written as a union of intervals
of the form (a; b) ), this means that K is neither open nor closed .
2