Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$
$M_2=\{(x,y) \mid x^2+y^2\le1\}$
What are the interior of $M_1$ and $M_2$ ?
And what are the boundary of $M_1$ and $M_2$ ?
How do I find them? Please show me so I can understand.
I know the answers but not the solutions.
Moreover – considering these examples – how can we say that topological boundaries are different from the geometric boundaries?
Please explain this in a clear and instructive way.
These are examples from my notebook which I need to understand well.
Thank you for help.
Best Answer
Interior (topological) $\operatorname{int} A$ of a subset $A$ of a topological space $X$ is the largest open set contained in $A$.
Topological boundary $\partial A$ of a subset $A$ of a topological space $X$ can be defined in various, equivalent ways, for instance as the difference $\overline A\setminus \operatorname{int} A$ between the closure of $A$ and its interior, or as the intersection $\overline A\cap \overline{A^c}$of closure of $A$ and closure of its complement, or more directly as the set of points whose every neighbourhood intersects $A$ and its complement.
Geometric boundary is something more inherent: when you have a manifold $M$ with boundary, it comes with an associated atlas of homeomorphisms (maps) $\varphi_\alpha:U_\alpha \to M$, where $U_\alpha$ is an open subset of $R^n_+=\{\overline x=(x_1,\ldots,x_n)\in {\bf R}^n\mid x_1\geq 0\}$ which agree in some way. Boundary $\partial M$ is defined then to be $\bigcup_\alpha \varphi_\alpha[U_\alpha\cap \{\overline x\mid x_1=0\}]$. $M\setminus \partial M$ can be referred to as the (geometric) interior of $M$.
For topological boundary and interior, $M_1$ is open and is in fact the interior of $M_2$ (because any ball containing points from the circle contains points from outside it), and you can use any of the two definitions to see that the boundary of both is the unit circle.
For geometric boundary and interior, with the natural manifold structure $M_1$ has no boundary (it is an open subset of ${\bf R}^2$), while $M_2$ has as its boundary, again, the unit circle; consequentially, geometric interior of both is $M_1$.