QN1: Consider the lower limit topology on $\mathbb{R}$ and the topology different from the lower limit topology given by the basis $\mathcal{B}=\{[a,b)|a < b,\, a\text{ and }b\text{ rational}\}$. Determine the closures of intervals $A=(0,\sqrt{2})$ and $B=(\sqrt{2},3)$ in these topologies.
QN2: Let $f\colon A \rightarrow B$ and $g\colon C \rightarrow D$ be continuous functions. Define a map $f\times g\colon A\times C \rightarrow B\times D$ by $(f\times g)(a\times c)=f(a)\times g(c)$.Show that $f\times g$ is continuous.
I think that the closure of $A$ in the lower limit topology is $[0,\sqrt{2})$ and the closure of $B$ in the lower limit topology is $[\sqrt{2},3)$. I would like to know the closures in the other topology.
For QN2 I think to attempt as follows.
By continuity of $f$ and $g$, the inverse images of $B$ and $D$ are open in $A$ and $C$, respectively. Then the product of these images should be open in $A\times C$. This is the way, I think, to deduce the continuity of $f\times g$, am I right?
Best Answer
Question 1:
(a) [Lower Limit Topology] You are correct that the closures of $(0,\sqrt{2})$ and $(\sqrt{2},3)$ in the lower limit topology are $[0,\sqrt{2})$ and $[\sqrt{2},3)$, respectively.
(b) [Other Topology] Hint: Prove that the closure of $(0,\sqrt{2})$ in this topology is $[0,\sqrt{2}]$; use the irrationality of $\sqrt{2}$. Can you see that the closure of $(\sqrt{2},3)$ in this topology is $[\sqrt{2},3)$?
Question 2:
Let us choose the "standard basis" for the product topology on $B\times D$ consisting of all sets of the form $U\times V$ where $U$ is open in $B$ and $V$ is open in $D$. We wish to establish the continuity of $f\times g$.
(a) Show that it suffices to prove the preimage under $f\times g$ of each element of this basis is open in $A\times C$. (Hint: use the definition of a basis.)
(b) Prove that if $U$ is open in $B$ and $V$ is open in $D$, then $(f\times g)^{-1}(U\times V)=f^{-1}(U)\times g^{-1}(V)$. Deduce that $f\times g$ is a continuous function.