Open and closed sets give us a way to define concepts such as nearness and connectedness without needing a metric space.
There are a couples of ways to introduce open and closed sets.
First Approach
Here we imagine that we have a device that measures distance imperfectly, it is just good enough to find which elements are adjacent to a given element.
Second Approach
Here we get rid of the concept of distance completely, instead we use subsets. The concept of 'nearness' is now just the elements in the same set.
Below we represent elements of sets as black dots, the sets are indicated by the red, green and blue lines, so if a black dot is inside the red line then it is a member of the red set.
Open Set These sets are open, they do not include their boundary. 

Closed Set These sets are closed, they include their boundary. 

Some sets are neither open or closed. 
Topology on a Set
A topology on a set X is a collection Τ of subsets of X, called open sets satisfying the following properties:
 X and Ø are elements of Τ.
 Τ is closed under finite intersections.
 Τ is closed under arbitary unions.
Open Set Criterion for Continuity
A map ƒ: M_{1} → M_{2} is continuous if and only if the inverse image of every open set is open.