Something basic is First Concepts of Topology, a positively reviewed book that is both cheap and short, and it also reads like a narrative (there are exercises if you want to enhance your understanding, but you obviously do not have to do them unless you want--full solutions to the exercises are available in the back of the book).
Perhaps more along the lines of what you are looking for is David Richeson's Euler's Gem: The Polyhedron Formula and the Birth of Topology, a very positively reviewed book (and winner of the 2010 Euler Prize by the MAA) where you do not have to do any exercises or constantly look up definitions. From the most helpful review from the link:
If you wanted a popularized book-length treatment of topology before Dave Richeson's Euler's Gem: The Polyhderal Formula and the Birth of Topology, you had no choice at all.
I'd highly recommend Euler's Gem, but it is longer than the first book and it is also more expensive; nonetheless, it is a book you could probably enjoy rather easily (i.e. it is something to read while relaxing and definitely something good to accompany a textbook if you want).
This post imported from StackExchange Mathematics at 2015-07-10 08:46 (UTC), posted by SE-user Daniel W. Farlow