I am trying to show that a certain space is a manifold. However, I am a bit unsure whether my current definition of compatible charts (Definition (1)) is similar to another definition of $C^{\infty}$-compatible charts (Definition (2)). and is it sufficient with my type of definition of compatible charts to show that a space is a manifold if we can exhibit a collection of compatible charts?
My definition goes as follows:
Definition (1) (M is any general set)
Two charts $(\mathcal{U}, \varphi)$ and $(\mathcal{V}, \psi)$ of $M$ are compatible if they have the same dimension $n$ and either $\mathcal{U} \cap \mathcal{V} = \emptyset$, or $\mathcal{U} \cap \mathcal{V} \neq \emptyset$ and:
$\varphi(\mathcal{U} \cap \mathcal{V})$ is open in $\mathbb{R}^n$
$\psi(\mathcal{U} \cap \mathcal{V})$ is open in $\mathbb{R}^n$ and
$\psi \circ \varphi^{-1} : \varphi(\mathcal{U} \cap \mathcal{V}) \rightarrow \psi(\mathcal{U} \cap \mathcal{V})$ is a diffeomorphism.
Definition (2)
Two charts $(\mathcal{U}, \varphi : U \rightarrow \mathbb{R}^n)$, $(\mathcal{V}, \psi : \mathcal{V} \rightarrow \mathbb{R}^n)$ of a topological manifold are $C^{\infty}$-compatible if the two maps $ \varphi \circ \psi^{-1} : \psi(\mathcal{U}, \cap \mathcal{V}) \rightarrow \varphi (\mathcal{U} \cap \mathcal{V}), \ \ \ \psi \circ \varphi^{-1} : \varphi(\mathcal{U} \cap \mathcal{V} \rightarrow \psi(\mathcal{U} \cap \mathcal{V})) $
are $C^{\infty}$
Note that $C^{\infty}$ in this case means smooth.
The definitions concern two different situations.
You do not give all details for Definition 1, but I guess that a chart $(\mathcal U,\varphi)$ of dimension $n$ on a set $M$ is a bijection from a subset $\mathcal U \subset M$ to an open subset $U'$ of $\mathbb R^n$. Since $M$ does not yet have a topology, it does not make sense to require that $U$ is open in $M$ or that $\varphi$ is a homeomorphism.
This is why you have the requirement that $\varphi(\mathcal{U} \cap \mathcal{V})$ and $\psi(\mathcal{U} \cap \mathcal{V})$ are open in $\mathbb{R}^n$; that cannot be deduced from the definition of a chart on the set $M$. Note that the transition function $\psi \circ \varphi^{-1} : \varphi(\mathcal{U} \cap \mathcal{V}) \rightarrow \psi(\mathcal{U} \cap \mathcal{V})$ is always a bijection, even if there are no assumptions on the openness of $\varphi(\mathcal{U} \cap \mathcal{V})$ and $\psi(\mathcal{U} \cap \mathcal{V})$. But the requirement that $\psi \circ \varphi^{-1}$ is a diffeomorphism in the standard sense of multivariable calculus needs the assumption that domain and range are open.
An atlas on set $M$ consisting of compatible charts allows to define a topology on $M$ such that all $\varphi$ become homeomorphism. This makes $M$ a topological manifold and the collection of compatible charts forms a smooth atlas on $M$.
Definition 2 deals with charts on a topological space $M$. A chart $(\mathcal U,\varphi)$ of dimension $n$ on $M$ is a homeomorphism from an open subset $\mathcal U \subset M$ to an open subset $U'$ of $\mathbb R^n$. In that case $\psi \circ \varphi^{-1}$ is automatically a homeomorphism between open subsets of $\mathbb R^n$ and we can require that it is smooth.
The requirement that both $\psi \circ \varphi^{-1}$ and $\varphi \circ \psi^{-1}$ are smooth is equivalent to the requirement that $\psi \circ \varphi^{-1}$ is a diffeomorphism.
An atlas on a topological manifold $M$ consisting of compatible charts forms a smooth atlas on $M$.