"Product" topology of $\mathbb N\times \mathbb R$ Background: Consider a set that is a finite or countable set (for example $\mathbb N$) endowed with discrete topology
Question: What topology is endowed with $\mathbb N\times \mathbb R$? Is $\mathbb N\times\mathbb R$ connected?
Is it possible to define a reasonable topology such that $\mathbb N\times\mathbb R$ is connected?
Motivation: "Connectness" is a useful property that is a necessary condition for many theorems. In order to apply those theorems in discrete space $\mathbb N$, one may want try to "connectify" the discrete space, for example, by "put it together" with $\mathbb R$.
My guess: $\mathbb N\times\mathbb R$ is not connected by its standard product topology. Consider set $\{1\}\times\mathbb R$ and $(\mathbb N\setminus \{1\})\times\mathbb R$, both of those sets are open by the standard product topology as $\mathbb R$ and $\{1\}$ are both open sets.
In order to make $\mathbb N\times\mathbb R$ connected, can we force define that the whole set $\mathbb R$ to be neither closed nor open? Intuitively, take a closed interval $[0,1)$, then $\mathbb N\times [0,1)$ can only be connected?
Additionally, I am not sure if things like $\mathbb N\times \mathbb R$ is connected. Any comments will help!
 A: The easiest way to make N×R connected is to give N the indiscrete topology.
A: Why forcing properties on the topology of $\mathbb R$ when $\mathbb R$ is already connected?
The obstruction in $\mathbb N\times \mathbb R$ connection is clearly the fact that $\mathbb N$ itself is not connected (which is what you used to prove that the product is not connected, i.e., being $\{1\}$ open in $\mathbb N$, you conclude that $\{1\}\times\mathbb R$ is open in $\mathbb N\times\mathbb R$), therefore I'd say that the easiest way to get to a connected topology on $\mathbb N\times \mathbb R$ is requiring a connected $\mathbb N$ (for instance, giving it the trivial topology, i.e. $\{\varnothing,\mathbb N\}$).
If your question requires us to keep the discrete topology on $\mathbb N$, I'm afraid that changing the topology of $\mathbb R$ won't be much of a help. You could always take a clopen set of $\mathbb N$ (e.g., $\{1\}$), and its product with any other topological space will still be a clopen in the product topology.
