Topology is different from topography. Informally, topology studies how elements of a set relates to each other spatially, in that sense, there is somewhat connections between topology and topography.

Mathematically, topology (from the Greek words τόπος, ‘place, location’, and λόγος, ‘study’) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topology space is a set endowed with a structure. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

In stead of thinking it as a geometrical problem, the gist of topology is algebra, thinking it of set theory or more specifically in math, functional analysis. The notion of homeomorphism is essential in Topology. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from “squishing” some larger object. (from wiki)

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

1. Both the empty set and X are elements of τ.
2. Any union of elements of τ is an element of τ.
3. Any intersection of finitely many elements of τ is an element of τ.

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X_τ may be used to denote a set X endowed with the particular topology τ. By definition, every topology is a π-system.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called a neighborhood of x.

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point.