Denote by $\mathbb{Z}$ the set of whole numbers, by $\mathbb{R}$ the set of real numbers, and by $\overline{\mathbb{R}}$ the set extended real numbers $\mathbb{R}\cup\{\pm\infty\}$. We denote intervals of extended real numbers as usual by $[a,b]$, $[a,b)$, $(a,b]$, and $(a,b)$. Define $\mathbb{N}_1 = \mathbb{Z}\cap[1,\infty)$, so that $\mathbb{N}_1 = \{1,2,\dots\}$,
Denote by $\mathbf{S}$ the set consisting of all the subsets of $\mathbb{R}^2$ of the form $[a,a+1]\times[b,b+1]$ for some $a, b \in \mathbb{Z}$. The members of $\mathbf{S}$ shall be referred to as squares. A chain in $\mathbf{T} \subseteq \mathbf{S}$ is a non-empty, finite sequence $S_1, S_2, \dots, S_n \in \mathbf{T}$, such that for every $k \in \{1,2,\dots,n-1\}$, $S_k$ and $S_{k+1}$ have at least one side in common. $A, B \in \mathbf{S}$ are said to be $\mathbf{T}$-connected iff there is a chain $S_1, S_2, \dots, S_n$ in $\mathbf{T}$ with $S_1 = A$, and $S_n = B$. It can be verified that the relation of $\mathbf{T}$-connectedness is an equivalence relation on $\mathbf{T}$. The corresponding equivalence classes shall be called the components of $\mathbf{T}$. $\mathbf{T}$ is said to be connected iff it has a single component.
If $T$ is a subset of the Euclidean plane, denote by $\overline{T}$ the topological closure of $T$. If $P$ is a Jordan curve in the Euclidean plane, denote by $i(P)$ the interior of $P$, and by $e(P)$ the exterior of $P$.
Let $\mathbf{T}$ be a non-empty, finite, connected subset of $\mathbf{S}$, and define $T = \cup\mathbf{T}$. I wish to show that $T$'s boundary can be represented as the union of $k \in \mathbb{N}_1$ closed, simple polygonal curves $P_1, \dots, P_k$, such that $T$ is contained in $\overline{i(P_1)}$, and for every $i \in \{2, \dots, k\}$, $T$ is contained in $\overline{e(P_i)}$.
It is easy to convince oneself of the validity of this claim when $\mathbf{T}$ consists of a single square. However, the general case eludes me.
Source
The claim I wish to prove is a major step in the proof of Theorem 4.14 "A bounded domain $G$ is simply connected if and only if whenever $G$ contains a closed Jordan curbve $\gamma$, $G$ also contains $I(\gamma)$" on pp. 70-72 of A. I. Markushevich's Theory of Functions of a Complex Variable, Three volumes in one, 2nd Edition, Chelsea Publishing Company, 1977.