I am trying to proof this statement. $f(x,y)=x ,\mathbb {R}^2 \rightarrow\mathbb {R}$ is mapping every open set to an open set, every closed set to a closed set and every copact set to a compact set.
firstly I think, that we proved somewhere in the lecture, that these kind of a map is continuous.So I can use this fact. I have a feeling , that all 3 statements are true, but I am confused by the question how exactly to show that. Should I try a general proof with the definitions, or are there any better ways? Thank you
Compact goes to compact by virtue of continuity of the projection. Open sets map to open sets because this holds for the basic open product sets. (in fact on any product space the projections are open and continuous).
It does not map closed sets to closed sets though: $C = \{(x,y): xy = 1 \}$ is closed in $\mathbb{R}^2$ but $f[C] = \mathbb{R}\setminus \{0\}$ is not closed in $\mathbb{R}$.
Projections $X \times Y \to X$ are closed maps precisely when $Y$ is compact, in fact. (Kuratowski's theorem).