I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology?
I know it's possible to learn topology without any knowledge of metric spaces. But I heard that knowing metric spaces will be good to motivate things.
Also , What resources do you recommend to gain this amount of knowledge of metric spaces ? I intend to learn metric spaces in the same time going through topology. Is that possible ? or the metric space needed is too much to be covered in the same time ?
I hope that your answers contain resources ( books , chapter of books , free notes on web , videos etc ... ) to learn this amount of metric spaces.
Note: I talk about point-set topology.
I cannot recommend this book enough: http://www.amazon.com/Introduction-Metric-Topological-Spaces-Mathematics/dp/019956308X
Generally one approaches metric spaces before general topology because metric spaces are in general more concrete/intuitive, and topology is somewhat more of an abstraction. Generally in order to understand concepts in topology, one typically refers to examples in Euclidean metric spaces, so a basic knowledge of such metric spaces is probably advisable before moving onto general topology.
The following playlist is also a good introduction to metric spaces/topology: http://www.youtube.com/playlist?list=PLF94A6F65866F3F31
If you want to keep waffle to a minimum, the following set of lecture notes form a solid foundation: http://www2.imperial.ac.uk/~svanstri/Files/ma222.pdf