In general topology by Wiilard, He mentioned in the exercises to the Looped line topology defines as follows: At each point $x$ of the real line other than the origin, the basic neighbourhoods of $x$ will be the usual open intervals centred at $x$. Basic neighbourhoods of the origin will be the sets: $(-\epsilon, \epsilon) \cup (-\infty, -n) \cup (n, \infty)$, for all possible choices $\epsilon > 0$ and $n \in \mathbb{N}$. If there is another that it mentions to Looped line topology?
Also, in, The looped line topology (Willard #4D), the user was asked to verify that the Looped line is a topology.
Done so far. I was able to see that it is $T_2$, compact, by using the definition. Also, it is metrizable since it is regular and second countable.
Interested in. I would like to see the reason why it is homomorphic to extend topology on the real line, $[-\infty, \infty].$
Attempt. I was trying to send $-\infty$ and $\infty$ to $0$ and send other points to themself but I could not finish.
Any help?