Topology
In many problems, we often divide a large space into smaller areas. For instance, a house is divided into rooms, a nation into states, a type of quantity into numbers, and so on. Each of these smaller areas (room, state, number) is next to other small areas (other rooms/states/numbers). The places where the areas meet are connections. If we write down on paper a list of spaces, and the connections between them, we have written down a description of a space -- a topological space. All topological spaces have the same properties such as connections, and are made of the same structure (a list of smaller areas). This makes it easier to study how spaces behave. It also makes it easier to write algorithms. For instance, to program a robot to navigate a house, we simply give it a list of rooms, the connections between each room (doors), and an algorithm that can work out which rooms to go through to reach any other room. For more examples of this type of problem, look at Graph theory.