I have never been interested in this before, and I have become interested in to find some answers and my teaching on the fundamental group has led me in this direction. Neither, I don't know if it suits for MO. So, please down-vote or vote-to-close after you provided some references.

I think if $X$ is semi-locally simply connected, simply connnected, and locally path connected then for the universal cover $p:\widetilde{X}\to X$, its group of Deck transformations, say $\mathrm{Deck}(p)$ acts properly discontinuously on $X$, hence it is equipped with the discrete topology. This then would imply that any covering over $X$ must have discrete group of Deck transformations. Indeed, this does not imply that it is finite, infinite or even countable. So, it could be $\mathbb{R}^\delta$ that is $\mathbb{R}$ equipped with the discrete topology. Is this conclusion correct?

So, if I wish to find a covering whose group of Deck transformations has a non-discrete topology then either $X$, the base space of my covering map, should not be semi-locally simply connected ( like the infinite earring) or $X$ should not be either path connected nor locally path connected. I wonder if there is a place that I can look for examples of such coverings where the topology of the Deck group is determined. I know that in Munkres's book there are some statements/exercises about this. But, anything more recent or some survey articles on this?

I have never been interested in this before, and I have become interested in to find some answers and my teaching on the fundamental group has led me in this direction. Neither, I don't know if it suits for MO. So, please down-vote or vote-to-close after you provided some references.

I think if $X$ is semi-locally simply connected, path connnected, and locally path connected then for the universal cover $p:\widetilde{X}\to X$, its group of Deck transformations, say $\mathrm{Deck}(p)$ acts properly discontinuously on $X$, hence it is equipped with the discrete topology. This then would imply that any covering over $X$ must have discrete group of Deck transformations. Indeed, this does not imply that it is finite, infinite or even countable. So, it could be $\mathbb{R}^\delta$ that is $\mathbb{R}$ equipped with the discrete topology. Is this conclusion correct?

So, if I wish to find a covering whose group of Deck transformations has a non-discrete topology then either $X$, the base space of my covering map, should not be semi-locally simply connected ( like the infinite earring) or $X$ should not be either path connected nor locally path connected. I wonder if there is a place that I can look for examples of such coverings where the topology of the Deck group is determined. I know that in Munkres's book there are some statements/exercises about this. But, anything more recent or some survey articles on this?

I have never been interested in this before, neither I don't know if it suits for MO. So, please down-vote or vote-to-close after you provided some references.

I think if $X$ is semi-locally simply connected, simply connnected, and locally path connected then for the universal cover $p:\widetilde{X}\to X$, its group of Deck transformations, say $\mathrm{Deck}(p)$ acts properly discontinuously on $X$, hence it is equipped with the discrete topology. This then would imply that any covering over $X$ must have discrete group of Deck transformations. Indeed, this does not imply that it is finite, infinite or even countable. So, it could be $\mathbb{R}^\delta$ that is $\mathbb{R}$ equipped with the discrete topology. Is this conclusion correct?

So, if I wish to find a covering whose group of Deck transformations has a non-discrete topology then either $X$, the base space of my covering map, should not be semi-locally simply connected ( like the infinite earring) or $X$ should not be either path connected nor locally path connected. I wonder if there is a place that I can look for examples of such coverings where the topology of the Deck group is determined. I know that in Munkres's book there are some statements/exercises about this. But, anything more recent or some survey articles on this?

I have never been interested in this before, and I have become interested in to find some answers and my teaching on the fundamental group has led me in this direction. Neither, I don't know if it suits for MO. So, please down-vote or vote-to-close after you provided some references.

I think if $X$ is semi-locally simply connected, simply connnected, and locally path connected then for the universal cover $p:\widetilde{X}\to X$, its group of Deck transformations, say $\mathrm{Deck}(p)$ acts properly discontinuously on $X$, hence it is equipped with the discrete topology. This then would imply that any covering over $X$ must have discrete group of Deck transformations. Indeed, this does not imply that it is finite, infinite or even countable. So, it could be $\mathbb{R}^\delta$ that is $\mathbb{R}$ equipped with the discrete topology. Is this conclusion correct?

So, if I wish to find a covering whose group of Deck transformations has a non-discrete topology then either $X$, the base space of my covering map, should not be semi-locally simply connected ( like the infinite earring) or $X$ should not be either path connected nor locally path connected. I wonder if there is a place that I can look for examples of such coverings where the topology of the Deck group is determined. I know that in Munkres's book there are some statements/exercises about this. But, anything more recent or some survey articles on this?