# discrete

This entry aims at highlighting the fact that all uses of the word discrete in mathematics are directly related to the core concept of discrete space:

• A discrete set is a set that, endowed with the topology implied by the context, is a \PMlinkescaptetextdiscrete space. For instance for a subset of $\mathbb{R}^{n}$ and without information suggesting otherwise, the topology on the set would be assumed the usual topology induced by norms on $\mathbb{R}^{n}$.

• A random variable $X$ is discrete if and only if its image space is a discrete set (which by what’s just been said means that the image is a discrete topological space for some topology specified by the context). The most common example by far is a random variable taking its values in a enumerated set (e.g. the values of a die, or a set of possible answers to a question in a survey).

• Discretization of ODEs and PDEs is the process of converting equations on functions on open sets of $\mathbb{R}^{n}$ (with boundary conditions) into equations on functions on discrete subsets of $\mathbb{R}^{n}$.

Title discrete Discrete 2013-03-22 17:56:49 2013-03-22 17:56:49 lalberti (18937) lalberti (18937) 8 lalberti (18937) Definition msc 54A05 Discrete