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Let $A\in {\mathbb{Z}}^2\times {\mathbb{Z}}^2$ be an expanding matrix, $𝒟\subset {\mathbb{Z}}^2$ a set with $|det(A\left)\right|$ elements and define $𝒯$ via the set equation $A𝒯=𝒯+𝒟$. If the two-dimensional Lebesgue measure of $𝒯$ is positive we call $𝒯$ a self-affine plane tile. In the present paper we are concerned with topological properties of $𝒯$. We show that the fundamental group ${\pi }_1\left(𝒯\right)$ of $𝒯$ is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of ${\pi }_1\left(𝒯\right)$. Furthermore, we give a short proof of the fact that the closure of each component of $\mathrm{int}\left(𝒯\right)$ is a locally...