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!
Best Answer
The easiest way to make N×R connected is to give N the indiscrete topology.