A topological space is a pair (X, U) where X is a set and U is a family of subsets (called the open sets) satisfying the following properties:
The family U of open sets is called a topology on X. A subset C in X is closed if X \setminus C is open.
X \times Y = \{(x, y) : x \in X, y \in Y\} for which the open sets are arbitrary unions of sets of the form U \times V, where U is an open set in X and V is an open set in Y.
Let X be any set. Let U = 2^X. This is a topology on X.
Let X be any set. Let U = \{\emptyset, X\}. This is a topology on X.
A topological space is Hausdorff if \forall_{p,q \in X, p \ne q}\exist_{U, V} p \in U \land q \in V \land U \cap V = \emptyset
Let f: X \to Y be a continuous bijection between subsets of Euclidean space. If X then f is a homeomorphism.
Let f,g : X \to Y be continuous maps. By a homotopy from f to g we mean a continuous map H : X \times [0; 1] \to Y such that H(x, 0) = f(x) and H(x, 1) = g(x).
If a homotopy exists then f and g are homotopic which is denoted by f \simeq g.
Given two topological spaces X and Y, a homotopy equivalence between those spaces are two continuous maps f: X \to Y and g: Y \to X such that f \circ g \simeq \text{id}_Y and g \circ f \simeq \text{id}_X. Then the two spaces are called homotopy equivalent.
Given (X, x_0) the fundamental group is
\pi_1(X, x_0) = [(S^1, 1), (X, x_0)]
Which is a group of the equivalence classes under the homotopy of loops based at x_0.
Let \sim be an equivalence relation on X. We define the quotient topology X / \sim to consist of all sets U in X / \sim for which \pi^{-1}(U) is open in X where \pi : X \to X/\sim is the quotient map given by x \mapsto [x]_\sim.