Continuous functions in topology I am trying to solve the following problem:
In which topologies (uniform, product, box) are the following functions from $\mathbb{R}$ to $\mathbb{R^\omega}$ continuous? 
$f(t)=(t,2t,3t,....)$
$g(t)=(t,t,t,.......)$
$h(t)= (t,\frac{1}{2}t, \frac{1}{3}t,...)$
I dont get why do the function have only one variable? and what is their meaning? Can we somehow draw them like calculus functions? It is not very intuitive. To prove continuity I think we have to prove that inverse images of these functions maps open sets to open sets in $\mathbb{R}$. How can we think of open sets here?
 A: $\Bbb R^\omega$ is the set of infinite sequences of real numbers; $f,g$, and $h$ are functions from $\Bbb R$ to this set of sequences. Take $f$, for instance: it sends the real number $\pi$ to the sequence $\langle \pi,2\pi,3\pi,\ldots\rangle$, meaning that $f(\pi)=\langle \pi,2\pi,3\pi,\ldots\rangle$. Similarly, $f(1)$ is the sequence $\langle 1,2,3,\ldots\rangle$ of positive integers, $f(-1)$ is the sequence $\langle -1,-2,-3,\ldots\rangle$, $f(0)=\langle 0,0,0,\ldots\rangle$, and
$$\begin{align*}
f\left(\frac16\right)&=\left\langle\frac16,\frac26,\frac36,\frac46,\frac56,\frac66,\frac76,\frac86,\ldots\right\rangle\\
&=\left\langle\frac16,\frac13,\frac12,\frac23,\frac56,1,\frac76,\frac43,\ldots\right\rangle\;.
\end{align*}$$
The function $g$ is even simpler: it just sends each real number to the constant sequence at that real number, so that $g(1)=\langle 1,1,1,\ldots\rangle$, $g(\pi)=\langle\pi,\pi,\pi,\ldots\rangle$, and so on.
As a final example, $h(\pi)$ is the sequence $$h(\pi)=\left\langle\pi,\frac{\pi}2,\frac{\pi}3,\frac{\pi}4,\ldots\right\rangle\;.$$
Three important topologies on this set of sequences are the uniform topology $\tau_u$, the product topology $\tau_p$, and the box topology $\tau_b$. Each of them makes $\Bbb R^\omega$ a topological space, and we can ask whether $f,g$, or $h$ is continuous when we give $\Bbb R^\omega$ one of these topologies and the domain $\Bbb R$ its usual topology. In order to answer the question, you’re going to have to have some understanding of the three topologies; knowing a base for each of them is sufficient. Suppose that $\mathscr{B}_u$ is a base for $\tau_u$, $\mathscr{B}_p$ is a base for $\tau_p$, and $\mathscr{B}_b$ is a base for $\tau_b$. To show, for instance, that $h$ is continuous with respect to $\tau_u$, you must show that

for any $x\in\Bbb R$ and any $B\in\mathscr{B}_u$ such that $h(x)\in B$, there is an $\epsilon>0$ such that $h(y)\in B$ whenever $|x-y|<\epsilon$,

thereby showing that $h$ is continuous at each $x\in X$.
To show that $h$ is not continuous with respect to $\tau_u$, you must

find an $x\in\Bbb R$ and a $B\in\mathscr{B}_u$ such that $h(x)\in B$, but for each $\epsilon>0$ there is a $y_\epsilon\in\Bbb R$ such that $|x-y|<\epsilon$ and $h(y)\notin B$:

this shows that $h$ is not continuous at $x$.
Your first step should be to identify bases $\mathscr{B}_u$, $\mathscr{B}_p$, and $\mathscr{B}_b$.
The ordinary product topology $\tau_p$ is normally described in terms of a particular basis; its elements are sets of the form $\prod_{k\in\Bbb Z^+}U_k$, where each $U_k$ is open in $\Bbb R$, and $U_k=\Bbb R$ for all but finitely many indices $k$.
The box topology is normally described in terms of a similar base: its elements are all products of the form $\prod_{k\in\Bbb Z^+}U_k$, where each $U_k$ is open in $\Bbb R$.
Finally, $\tau_u$ is usually described in terms of a nbhd base at each point. (Remember that a point in this space is actually an infinite sequence of real numbers.) For $x=\langle x_1,x_2,x_3,\ldots\rangle\in\Bbb R^\omega$ and $r>0$ let
$$N(x,r)=\{\langle y_1,y_2,y_3,\ldots\rangle\in\Bbb R^\omega:|x_k-y_k|<r\text{ for all }k\in\Bbb Z^+\}\;,$$
and let
$$B(x,r)=\bigcup_{0<s<r}N(x,s)\;;$$
then $\mathscr{B}_u=\left\{B(x,r):x\in\Bbb R^\omega\text{ and }r>0\right\}$ is a base for the uniform topology $\tau_u$ on $\Bbb R^\omega$.
If you’ve been given a slightly different definition of any of these topologies, you should try to see why it’s equivalent to what I’ve given here.
There isn’t any really good way to draw pictures of the maps $f,g$, and $h$.
