I really tried to understand the concept of open and closed set and boundary, but I did not get it quite well. I have questions about it. First is there anyone who tell me exactly these concepts?
And then I encounter a question about it like this:
$$S = \{(x, y)|x \in [0, 2] , 2x + y < 1\}\text{ where }x \in\mathbb R\text{ and }y \in\mathbb R\text,$$
About the nature of this set, how can I show that it is not bounded? and it is not closed? Is it open or neither open nor closed?
Also, how can I maximize $f(x, y) = x + 2y$ over the set $S$. Is it a problem that this set is not bounded and not closed?
Best Answer
Actually you could draw $y=1-2x$ and the below part of the area intersected with the rectangle $[0,2]\times \mathbb{R}$ it's your desired set (kind of infinite trapezium, with left and right borders being part of $S$ and up and down basis being not part of $S$ in turn $S$ is not open nor close). A way to see if a set it's closed it's to check the boundary, it contains the boundary or not? In this case it's just enough to check at $(0,1)$, this point it's on the boundary because for any small ball you get values on $S$ and outside $S$ i.e., for $x=0$ values such that $y<1$ and $y\geq 1$, and evidently $(0,1)\notin S$, so $S$ couldn't be closed. Try at the point $(0,0)$ and check that is on the boundary and it's on $S$ so $S$ can't be open.
Also, since $y$ is not bounded as the only condition it's $y<1-2x$ and $y\in \mathbb{R}$ we see that $-\infty<y<1$ therefore $S$ is not bounded.
The function $f(x,y)=x+2y$ has a local max, which is $(0,1)$ but is not on $S$ so $f$ couldn't be optimized on $S$.