TITLE: Topology for sets in $\mathbb{R^2}$ and $\mathbb{R^3}$
QUESTION [0 upvotes]: I have a question regarding topology of sets in $\mathbb{R^2}$ and $\mathbb{R^3}$. The question is as follows:
For each $A \subset \mathbb{R^3}$, we define $P(A) \subset \mathbb{R^2}$ by \begin{equation} P(A) := \{(x,y)\  |\ \exists\ z \in \mathbb{R}: (x,y,z) \in A\}.\end{equation}
I have the following statements: 
"If A is open (in $\mathbb{R^3}$), then $P(A)$ is open (in $\mathbb{R^2}$)." 
"If A is closed (in $\mathbb{R^3}$), then $P(A)$ is closed (in $\mathbb{R^2}$)."
"If A is compact (in $\mathbb{R^3}$), then $P(A)$ is compact (in $\mathbb{R^2}$)."
I think that the first statement is true, but I am in doubt about the second one. I was thinking of a set $A = \{(x,y,z)\  | \ x^2 + y^2 < 1, -1 \leq z \leq 1\}$, but I am now sure whether I am going in the correct direction. Could anyone help me out? Thanks in advance!

REPLY [0 votes]: The general def'n of the Tychonoff-product topology on $S(J)=\prod_{j\in J}S_j$ when each $S_j$ is a top'l space is that it is the weakest  topology such that  for each $a\in J$ the projection $p_a(\;(x_j)_{j\in J}\;)=x_a$ is continuous. 
Any $A\subset S(J)$  is open in this topology iff the following holds: For each $x=(x_j)_{j\in J}\in A$, there exists $A(x)=\prod_{j\in J}B_j \subset A$ such that (I)... $x\in A(x)$, and (II)... each $A_j$ is open in $S_j$ and  (III)... $\{j\in J: A_j\ne S_j\}$ is finite.
It follows that if $K\subset J$ and $P_K(\;(x_j)_{j\in J}\;)=(x_k)_{k\in K}$ then $P_K$ is a continuous open mapping from $S(J)$ to $S(K)=\prod_{k\in K}S_k.$ (A function is called an open mapping iff the image of each open set is open.)
In particular, when $J$ is finite then $A$ is open  in $S(J)$ iff for each $x=(x_j)_{j\in J}\in A$ there exists $A(x)=\prod_{j\in J}A_j\subset A$ with $x\in A(x)$, where each $A_j$ is open in $S_j.$ And hence  if $K\subset J$ then the image $P_K(A)$ satisfies $P_K(x)\in \prod_{k\in K}A_k\subset P_K(A),$ so $P_K(A)$ is open in $S(K)$.
For example, in your Q we have $J=\{1,2,3\}$ and $K=\{1,2\}$ and each $S_j=\mathbb R.$  Let $A$ be open in $S_J=\mathbb R^3.$ For any $(x,y,z)\in A\subset S(J)$  there exist open subsets $A_1,A_2,A_3$ of $\mathbb R$ such that $(x,y,z)\in A_1\times A_2\times A_3\subset A .$  So $(x,y)\in A_1\times A_2\subset P_K(A)$. This holds for every $(x,y,z)\in A$  so $P_K(A)$ is open in $S(K)=\mathbb R^2.$
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