Subsets in half space

Question

i'm studying manifolds with boundary but i'm stuck because i don't have idea how to define: If $A \subset H^n$ $A$ is open in $H^n$ Int $A$ $\partial A$

Anyone give me a hand please.

Answer 1

If you are asking for definitions, then $H^n$ the closed $n$-dimensional upper half space is defined as follows. $H^n = \{ x = \langle x_1, ..., x_n \rangle\in \mathbb{R}^n \ | \ x_n \geq 0\} \subseteq \mathbb{R}^n$

It is clear from the definition of $H^n$ that it inherits the subspace topology from $\mathbb{R}^n$. So if $A \subseteq H^n$, then $A$ is open in $H^n$ if $A = U \cap H^n$ for some $U$ open in $\mathbb{R}^n$.

Some basic facts about $H^n$, we have $\text{Bd}( H^n )= \{x = \langle x_1, ..., x_n \rangle\in \mathbb{R}^n \ | \ x_n = 0 \} \subseteq \mathbb{R}^n$, and $\text{Int}(H^n) = \{x = \langle x_1, ..., x_n \rangle\in \mathbb{R}^n \ | \ x_n > 0 \} \subseteq \mathbb{R}^n$. (Note I am talking about topologial boundary here)

The whole reason we define $H^n$ is that they help us to define manifolds with boundary. The best reference I have for this, which I can recommend is Introduction to Topological Manifolds by John Lee, on pages $42$ and $43$.