Reedeegi's list is pretty comprehensive, but I'll offer two recommendations:
1. Kolmogorov and Fomin's Functional Analysis is great if you know some basic analysis. The chapter on metric spaces (the chapter after the conventional first chapter on set theory in many undergraduate texts) is superb, as the authors don't hesitate to present some of the most important metrics in analysis with great detail. A lot of the basic topology needed for analysis is also introduced, and the material on normed linear spaces looks good (I'm on this part now). The last couple of chapters are on measure and integration, so the text covers both functional analysis and modern real analysis.
2. Counterexamples in Topology. This text helped me learn the general topology needed for my current analysis course. The very short and condensed introduction covers the basic facts on general topology, and quickly moves onto the various examples of topological spaces. While a lot of important theorems such as those due to Tychonoff or Urysohn aren't proven here, you'll learn a lot about topological properties by studying the examples (and counterexamples are pretty handy). Ideally, you would want to supplement a general topology text with this one, but I find that the introduction along with wikipedia suffices to understand most of the examples in the text.