Useful examples and counterexamples

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Useful examples and counterexamples

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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, $\mathbb{R}$, $\mathbb{R}^2$, $C[0,1]$, $\ell_\infty$, the unit ball of $\ell_\infty$, 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 $\mathbb{N}^{\mathbb{N}}$, the Stone-Cech compactification of $\mathbb{N}$, 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.

[[Examples of functions defined on the complex numbers]]

[[Examples of functions defined on the real numbers]]

[[Some useful examples of graphs]]

[[Basic examples of groups]]

[[Some examples of manifolds]]

[[Examples and counterexamples in metric spaces]]

[[Some important classes of polynomials]]

[[Some interesting sets of integers]]

[[Some interesting sets of real numbers]]

[[Examples of rings]]

[[Some important solutions of differential equations]]

[[Examples and counterexamples in topological spaces]]

[note article incomplete]
If you want to add a suggested article title and not-yet-operating link to this list then please do. And if you can actually write a first draft or even partial first draft of the article, then all the better. But bear in mind that just listing examples isn't very informative -- you should tell the reader what kinds of properties the examples have, so that others understand why they are useful. Similarly, if you are tempted to write about a counterexample to some conjecture, think first about your reasons for doing so. Some counterexamples are worth knowing about because they are counterexamples to plausible statements that people may waste time trying to prove -- not that that process is ever a complete waste of time -- or they can be adapted to disprove many different statements. Others are constructed in interesting ways that illustrate useful techniques: these are worth writing about in the Tricki, but probably not in a subpage of this one and instead in a page about the useful techniques themselves. Still others are very specific solutions to single problems: if you cannot extract any general moral from a counterexample then it is questionable whether it belongs in the Tricki.[/note]

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