how to learn mathematical topology?
It's all so dry and boring.
I don't want to do any of the examples.

How would you compare set theory and topology? Would you say they both have a similar role -- that being a basic language to communicate maths in?
Then why was set theory so easy, and topology so hard?

>>16715821 (OP)
Is there non-mathematical topology?

Jokes aside, start with differential geometry. You will naturally get into why you need topology along the way. For example, defining integration on manifolds requires partitions of unity, which require paracompactness, which is satisfied by Hausdorffness and second-countability yata-yata.
>How would you compare set theory and topology? Would you say they both have a similar role
No, they play two different roles. Set theory serves as the “playground” for both topology and algebra.
>Then why was set theory so easy, and topology so hard?
Because the category of sets has tons of “nice” properties (balanced, regular, cartesian closed, Barr-exact, bicomplete, total, has a power object, etc etc), whereas the category of topological spaces isn’t. There are “nice” topological spaces out there that satisfy these properties (eg compact Hausdorff spaces), but they have limited scope.

Learning mathematics is hard work and requires prolonged concentration on things that seem both boring and difficult. You will have to work through problems and examples and spend time breaking concepts down into understandable parts.