V. A. Vassiliev explains that his Introduction to Topology contains the lecture notes for a course he has taught several times at the Independent University of Moscow. It must have been quite a course. In little over 140 pages, the book goes all the way from the definition of a topological space to homology and cohomology theory, Morse theory, Poincaré duality, and more. The book emphasizes intuitive arguments whenever possible, but it also assumes a lot. For example, it assumes that the reader has a good algebra background and is willing to fill in a lot of the detail.
To make this more concrete, here is what is on pages 1 and 2: a short introduction, the definition of a topology, the discrete topology and the topology on a metric space as examples of topologies, the definition of a basis of a topology, an example of an "exotic" topology on R, the definition of closed sets and of the closure of a set, and an exercise: "Consider the basis consisting of parallelepipeds in Rn with edges parallel to the coordinate axes. Is the topology it determines a different one?"
Though this is part of the AMS Student Mathematical Library, it's hard to imagine an undergraduate who can learn topology from this book. On the other hand, I can imagine an advanced undergraduate enjoying the book as a broad survey of the field. It is often useful to have an overall picture of a subject before engaging it in detail. For that, this book would be a good choice.
Fernando Q. Gouvêa (firstname.lastname@example.org) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.