Topology is referred to as the mathematics of continuity or theory of abstract topological spaces. The concept of topological space grew out of the study of real line and euclidean space and the study of continuous function on these spaces.
One of the most important and frequently used ways of imposing topology on a set is to define topology in terms of metric on a set.In this page firstly we will discuss metric spaces, open sets, and closed sets to understand topology. Definition of topology will also give us a more generalized notion of the meaning of open and closed sets.

Topology Terms

Metric Space: A metric space is a set X where we have a notion of distance. If x, y $\epsilon$ + X, then d(x, y) is the distance between x and y.
For every x, y, z $\epsilon$ X it must satisfy the following properties.
• d(x, y) $\geq$ 0 $\forall$ x, y $\epsilon$ X
• d(x, y) = d(y, x) $\forall$ x, y $\epsilon$ X (symmetry)
• d(x, y) + d(y, x) $\geq$ d(x, z) $\forall$ x, y, z $\epsilon$ X
where the real number d(x, y) is often called distance between x and y in the metric 'd'.

Topological Space: Let X be a non empty set. A class $\tau$ of the subsets of X is a topology on X if and only if $\tau$ satisfies the following axioms.
• X and $\phi$ belong to $\tau$
• The arbitrary union of elements of $\tau$ is contained in $\tau$.
• Intersection of any two sets in $\tau$ belongs to $\tau$.
The elements of $\tau$ are called $\tau$ open sets and X together with $\tau$.
(X, $\tau$) is called a topological space.
Open sets: The set A is said to be open if and only if all the points of the set are contained in it.
Closed sets: Let X be a topological space. A subset A of X is a closed set if and only if its complement A$^{c}$ is an open set.
Closure of a set: Let A be a subset of a topological space X. The closure of A denoted by $\bar{A}$ is the intersection of all closed super sets of A.
Set A is closed if and only if  A = $\bar{A}$.
Dense sets: A subset A of a topological space X is said to be dense in B C X if B is contained in closure of A.
A is dense in X iff $\bar{A}$ = X
Nowhere dense sets: A subset A of a topological space X is said to be a nowhere dense in X if the interior of the closure of A is empty.
int($\bar{A}$) = $\phi$

Topology Examples

Example 1: Let T be the class of subsets of R consisting of set of rationals and irrationals and all open infinite interval of the form E$_{a}$= (a, $\infty$) where a $\epsilon$ R. Show that T is a topology on R.

Solution: Let A be a subclass of T where

A = {E$_{i}$, i $\epsilon$ I}

where I is same set of real number.

If I is not bounded below, Inf(I) = -$\infty$

Therefore  $\bigcup$E$_{i}$ = (-$\infty$, $\infty$) = R $\epsilon$ T

If I is bounded below

Inf(I) = i$_{0}$

$\bigcup$E$_{i}$ = (i$_{0}$, $\infty$) $\epsilon$ T

$\bigcup$E$_{i}$ $\epsilon$ T

Therefore, arbitrary union is satisfied.

Further finite intersection of all the sets in R is again in R.

Therefore T is a topology on R.

Example 2: If A C B then prove that A' C B'

Solution: Let p $\epsilon$ A'

$\rightarrow$  G - {p} $\bigcap$ A $\neq$ $\phi$  $\forall$ p $\epsilon$ G

G - {p} $\bigcap$ A C B $\neq$ $\phi$

Therefore G - {p} $\bigcap$ B $\neq$ $\phi$

$\rightarrow$ p $\epsilon$ B'

Therefore A' C B'

Example 3: Let A be any subset of a discrete topological space X, show that the derived set A' = $\phi$

Solution: Let p be an arbitrary point in X

Since every subset of discrete space in X is p

Therefore, {p} is an open set

Let G = {p}

[G - {p}] $\bigcap$ A = $\phi$

p is not a limit point of A

Since p is arbitrary point

Every point in X is not a limit point in X

Therefore A' =  $\phi$