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.

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    $\begingroup$ Note that you can get motivation not only from analysis and the theory of metric spaces, but also from commutative algebra, graph theory, category theory and many more areas of modern mathematics. General topology is used almost everywhere, these days. $\endgroup$ – Jakob Werner Jul 4 '14 at 23:54
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    $\begingroup$ Good question. I am late to catch it. +) $\endgroup$ – Mikasa Aug 3 '14 at 15:29
  • $\begingroup$ @BabakS. , You are never late :) I have already started topology since a while and learned some things about metric spaces. In fact, introducing the concepts without being a natural generalization of the properities of metric spaces was Not that bad as I thought before :) $\endgroup$ – Fawzy Hegab Aug 3 '14 at 15:35
  • $\begingroup$ @MathsLover: Good to hear that. :-) $\endgroup$ – Mikasa Aug 3 '14 at 15:36

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

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    $\begingroup$ I think another very good reason for studying metric spaces beforehand is that they are fairly well-behaved. Topological spaces can be very pathological and difficult to intuit. Most topological spaces we are interested in will end up, in some sense, being similar in nature to metric spaces anyway. $\endgroup$ – Cameron Williams Jul 5 '14 at 0:41
  • $\begingroup$ @Edward ffitch , What do you mean that "I can't recommend this book enough' ? Do u recommend it or not ? $\endgroup$ – Fawzy Hegab Jul 5 '14 at 0:54
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    $\begingroup$ I highly recommend it! $\endgroup$ – Edward ffitch Jul 5 '14 at 0:56

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