Definition. Diffeological space [math-001D]

A diffeological space is a pair $(X, \mathcal{D}_X)$ consists of a given set $X$ and a diffeology $\mathcal{D}_X$ consists of a collection of parameterizations $p : U \to X$ satisfying the following conditions:

1. All parameterizations with domain $\mathbb{R}^0$ belong to $\mathcal{D}_X$ , namely all the points of $X$
2. If $p : V \to X$ is a parameterization, and $f : U \to V$ is a smooth map between cartesian spaces, then $p \circ f$ belongs to $\mathcal{D}_X$
3. If $p : U \to X$ is a parameterization, an open cover $(U_i)_{i\in I}$ of $U$ that each restriction $p \mid_{U_i} \in \mathcal{D}_X$ , then $p \in \mathcal{D}_X$

If $\mathcal{D}_X$ is a diffeology, then we call a parameterization $p$ that belongs to it a plot.