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I am a mathematician who never had a course in logic, and I'd like to learn about it, especially modal logic and the relationship with topology. Can you recommend me an introductory book?

Feel free to ask me any questions that may be relevant to choose the recomendations.

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    $\begingroup$ Others will surely have more focused suggestions, but possibly you may want to see if The Mathematics of Metamathematics by Helena Rasiowa could be of any interest to you. I don't think it's online, at least freely available if it is, but you can find it in quite a few university libraries. $\endgroup$
    – Dave L. Renfro
    Commented Aug 10 at 22:05
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    $\begingroup$ I think there is a Curry-Howard correspondence between topological closure operators and modal logic modalities. For example, if $\square S$ represents the topological interior of $S$ we have $\square S\subset S$ and $\square S \subset\square\square S$ but not in general $S\subset \square S$. Analogously if $\square$ is the “necessarily” modality we have $\square S\to S$ and $\square S\to\square\square S$ but not in general $S\to\square S$. $\endgroup$
    – MJD
    Commented Aug 11 at 2:26
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    $\begingroup$ A nice overview is given by Bezhanishvili in his presentation Topological Semantics of Modal Logic. $\endgroup$
    – Tankut Beygu
    Commented Aug 11 at 9:15

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To begin, the book "Introduction to Modal Logic" by Blackburn, De Rijke, and Venema is a great resource for modal logic. It is a good place to start learning modal logic and to get exposure to more advanced topics and classic results. Here is a link to the publisher.

https://www.cambridge.org/core/books/modal-logic/F7CDB0A265026BF05EAD1091A47FCF5B

This being said, the connection to topology is not really discussed in the aforementioned book. As was already suggested by Tankut Beygu in the comments, these slides by Nick Bezhanishvili are a good place to start:

https://rodrigonalmeida.github.io/projects/Topology_Project/Guest-lecture-slides.pdf

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Rob Goldblatt's paper on the history of modal logic isn't what you are looking for because it won't teach you modal logic. But it does discuss the topological connections in some breadth and provides many references you could follow up.

You could learn the basic modal logic in many places, then use the Goldblatt paper to get the topological connections.

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