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.

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.