My favorite equation would have to be...
The definition of the discrete metric on a nonempty space X. The discrete metric is 0 if two inputs in X are the same and 1 if the two inputs are different. This IS an equation, as it is a function definition, where the function maps the Cartesian product of X with itself into the real line. The discrete topology, which is the topology induced by the discrete metric, is the largest possible topology on a space in terms of set inclusion-- that is, the discrete topology contains all other possibie topologies on a space.
Other fun facts about the discrete topology:
The discrete topology is Hausdorff; that is, any two distinct points in the space can be separated by disjoint open neighborhoods in the discrete topology, as singleton sets are open in the discrete topology.
A topological space with the discrete topology is second countable if and only if it is countable. If a metric space is not second countable, then it is not separable. Thus, since the real line with the discrete topology is uncountable, it is not second countable and is thus an example of a metric space which is not separable.
The identity function mapping the real line with the discrete topology onto the real line with the Euclidean topology is an example of a function which is bijective and continuous, but not bicontinuous and thus not a homeomorphism.
The subspace topology on the integers as a subspace of the real line with the Euclidean topology is the discrete topology.
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