Understanding the definition of boundary points of a manifold

Question

Let $M$ be a topological manifold. We call it a $n$-manifold with boundary if for each $x\in M$, there is a chart $(U,\phi)$ at $x$ such that $\phi$ is a homeomorphism from $U$ to an open subset of $\mathbb{H}^n$, where $\mathbb{H}^n=\{(x_1,x_2,\cdots, x_n): x_n\ge 0\}$. Define $x\in M$ to be a boundary point if $\phi(x)\in\partial\mathbb{H}^n$ for some chart $(U,\phi)$ at $x$. How can I show that this definition does not depend on the choice of the chart?