Many mathematical problems become much easier to solve if one has in mind a stock of important examples. For instance, if you are trying to do an exercise about metric or topological spaces, and if the exercise is of the form "Does there exist a metric/topological space such that ...?" then it is often possible to answer the question by checking the property in question against examples such as open intervals, closed intervals, , , , , the unit ball of , the discrete topology on an infinite set, the indiscrete topology on an infinite set, the cocountable topology on an uncountable set, the product topology on , the Stone-Cech compactification of , and so on. This is far from a complete list: for a longer one see Examples and counterexamples in metric spaces and Examples and counterexamples in topological spaces.
Similar articles have been written, or with luck will be written, for other areas of mathematics. They will not solve all your existence problems, but they are a good first port of call if you want to do a preliminary test of the truth or otherwise of a mathematical statement. This page is far from complete: see below for suggestions about how to contribute to it.