**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($\bar{A}$) = $\phi$