Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level.
I am wondering if there is a book or a set of books on topology like Rudin's analysis books, S.Lang's algebra, Halmos' measure (or to be more updated, Bogachev's measure theory), etc, serving as standard references.
Probably, such a book should hold characteristics such as being self-contained, covering the most of classical results, and other good properties you can name.
In the end, I only have the interest on general-topology (topological space, metrization, compactification...), and optionally differential topology (manifolds). So please don't divert into algebraic context.
Thanks for all the delightful replies. I will try to quickly check the books mentioned beneath. And I may accept the answer which is most closed to my personal flavor.
Sorry for others. It's a pity no multiple acceptance can be made for such a ref-request question.