Topologies on n-manifolds In the study of n-manifolds (real and imagined), is there any reason to spend much time learning about topologies other than the usual topology?
 A: Here is my interpretation of your question: Do I ever need to consider topological spaces which are "far" from being manifolds, if I am interested in manifold topology?  
First, to study topology of manifolds you need to have a solid general topology (aka point set topology) background. It does not mean that you will ever need to use, say, the lower limit topology or the long line topology, but you need to be comfortable with most topological concepts, especially subspace topology, quotient topology, Hausdorffness, compactness, connectedness, continuity, homeomorphism, Baire category. (All in abstract topological setting.) You also might need to known the Lebesgue covering dimension (especially if you study topological manifolds without any smoothness assumption). Occasionally, you might need higher separation properties (regular and normal spaces), typically in the context of Urysohn-Titze extension theorem. 
Most topologies you see in the context of manifolds are (unsurprisingly) locally compact, Hausdorff, 2nd countable (since they are typically subspace topologies on subsets in a topological manifold). However, there are exceptions as sometimes it is useful to think about non-Hausdorff topologies (e.g. in theory of foliations on manifolds; in connection to algebraic geometry - Zariski topology). Also, sometimes (or often, depending on what you do) you might need topologies on spaces of maps between manifolds (e.g. compact-open topology, Frechet topology, etc), as well as topology on the set of closed subsets of the given manifold. If you are planning to do topological gauge theory, you absolutely have to have a strong analysis background, which requires you to be comfortable with Banach spaces (including Sobolev spaces). 
