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I understand the notion of converging to a point for a sequence in a general topological space, and also understand its coincidence with that of convergence of sequences of real numbers. My problem is I can't figure out a similar definition for the case converging to the infinity in general topological spaces! I have an idea that the open sets in a topology measure how close a point is relative to another point. But I don't understand how a topology expresses infinity or far away in terms of open sets.

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    You might want to read about "ends" in topology. – 2017-01-13
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    @JohnHughes Thanks. I checked out the [wiki](https://en.wikipedia.org/wiki/End_(topology)) and I guess this is the answer although it seems more complicated than I expected, compared with the notion of the convergence to a point :-) – 2017-01-13
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    That's probably because it *is* more complicated. Life's like that sometimes... :) I've added my comment as a community-wiki answer, so that you can accept it and this question can be put to rest. – 2017-01-13

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You might want to read about "ends" in topology.

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You can use Alexandroff compactifications of your space X. Then a sequence $(x_n)$ in X diverges to infinity iff it eventually escapes every compact subset of X: $x_n\to\infty$ iff for each compact $K\subset X$ there exists $N$ such that $\forall n>N, x_n\notin K$.

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    Indeed this is a good idea to express converging to infinity, because it looks very similar to the case of converging to a point. – 2017-01-15