Topological space

In real analysis, a topology is an informal way to relate how objects in a set – say $$X$$ – relate to one another spatially. Members of a topology can be described as open sets of $$X$$, similar to how open balls play a significant role in the construction of a metric. In a sense, topologies are an abstraction of a metric and thus, every metric induces a topology (but not necessarily the converse).

Definition
Let $$X$$ be a set and let $$\tau$$ be a family of subsets in $$X$$. We say that $$\tau$$ is a topology on $$X$$ if Additionally, we say that the tuple $$(X, \tau)$$ is a topological space.
 * $$\varnothing, X$$ are elements of $$\tau$$;
 * The union of elements in $$\tau$$ are also elements in $$\tau$$;
 * The finite intersection of elements in $$\tau$$ are also elements in $$\tau$$.

Coarse topology
The most natural example of a topology is the coarse topology (also called the trivial topology). In a sense, this is the simplest form of a topology. We say that $$\tau$$ is the coarse topology on the set $$X$$ if $$\tau = \{\varnothing, X\}$$. It is easy to check that all three axioms of a topology hold.

Discrete topology
The next example of a topology is the discrete topology. Every topology $$\tau$$ is a subset of the power set of $$X$$. The discrete topology is the topology on $$X$$ if $$\tau = \mathcal P(X)$$ where $$\mathcal P(\cdot)$$ describes the power set. The discrete topology on a set $$X$$ is the metric topology of the discrete metric on $$X$$.

Hausdorff topological space
We say that the topological space $$(X, \tau)$$ is Hausdorff if, for any two distinct points $$x, y \in X$$, there exist neighborhoods which are disjoint from each other. That is, any two distinct points in $$X$$ can be separated by distinct neighborhoods. Every metric space is Hausdorff, so every metric topology is a Hausdorff space.