\documentclass{article}

\usepackage{bbm}
\usepackage{amsmath}

\newcommand{\N}{\mathbbm{N}}

\newtheorem{Def}{Definition}
\newtheorem{The}{Theorem}[Def]

\DeclareMathOperator{\Dom}{Dom}
\newcommand{\Dim}[1]{\tilde{#1}}

\begin{document}

Note: we assume here that dimensions are cells instead of strings.

\begin{Def}[space]
A ZigZag space is the triplet $Z = (X, c, C)$ where $X$ is the set of
cells, $c : X \rightarrow \Sigma^*$ is the cell content mapping
and $C \subset X^3$ is the set of connections, if the following hold:
%
\begin{enumerate}
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\item If $(d, e, f) \in C$ and $(d, e', f) \in C$ then $e = e'$.
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\item If $(d, e, f) \in C$ and $(d, e, f') \in C$ then $f = f'$.
%
\end{enumerate}
\end{Def}

\begin{Def}[subspace]
A ZigZag space $Z' = (X', c', C')$ is a subspace of another ZigZag
space $Z = (X, c, C)$ if the following hold:
%
\begin{enumerate}
%
\item $X' \subset X$
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\item $c'(x) = c(x)$ for every $x \in X'$.
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\item $C' \subset C$
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\end{enumerate}
\end{Def}

\begin{The}
  Let $Z = (X, c, C)$ be a ZigZag space.  For every $d \in X$, there
  exists a maximal $X' \subset X$ and a unique $\Dim{d} : X'
  \rightarrow X$ such that for every $x \in X'$ it holds that $(d, x,
  \Dim{d}(x)) \in C$.  Additionally, $\Dim{d}$ is an injection.
\end{The}

\begin{Def}[clone dimension]
  Let $Z = (X, c, C)$ be a ZigZag space.  We say that $d \in X$ is a
  clone dimension if, for every $x \in \Dom d$ it holds that $c(x) =
  c(\Dim{d}(x))$.
\end{Def}

\end{document}