I have to prove that $Q=\{(x_1,...,x_n)\in \mathbb R^n\mid \forall i=1,...,n,\ x_i\geq 0\}$ is a topological manifold with boundary. The fact that the topology is second countable and hausdorff is a consequence of the fact that it's a subset of $\mathbb R^n$ and thus it inherit of those properties from $\mathbb R^n$.
1) Is it correct to say that ?
I have difficulties to show that it's locally homeomorphic to $\mathbb H^n=\{(x_1,...,x_n)\mid x_n\geq 0\}$, any idea ?
To answer your first question: Yes, any subset of $\mathbb R^n$ (with the subspace topology) is Hausdorff and second countable, because those properties are hereditary.
Suggestion: To show that $Q$ is a topological manifold with boundary, first prove that if $f\colon \mathbb R^{n-1}\to \mathbb R$ is any continuous function, then the set $$\{(x_1,\dots,x_n)\in\mathbb R^n: x_n\ge f(x_1,\dots,x_{n-1}\}$$ is a topological manifold with boundary. Then try to find a linear isomorphism from $\mathbb R^n$ to itself that takes $Q$ to a set of this form.
