Good reference for metric topology I would like a "good" book (not really introductory, not too advanced with good theory and exercises) on metric topology covering the following topics:

Metric spaces, open/closed sets, sequences, compactness, completeness, continuous functions and homeomorphisms, connectedness, product spaces, Baire category theorem, completeness of C[0, 1] and Lp spaces, Arzela-Ascoli theorem.

It may not be a full book but parts of books or lecture notes are also most welcome.

Chapter 4: Metric and metrizable spaces.... in "General Topology" , by R. Engelking. You need to browse the Introduction to familiarize yourself with his notation (e.g. $\{x_i\}$ is a sequence), and be aware that for him, compact space means compact Hausdorff.... Chapter 3 is Compact spaces. It includes a beautiful short proof of the Stone-Weierstrass Theorem ( see 3.2.18 thru 3.2.21).... There is a wealth of extra material in the problems and exercises of this book.... But for $C[0,1], L^p,$ and Arzela-Ascoli, etc., I would suggest a text on functional analysis.  
