Cr topology

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In mathematics, the C^r topology is the topology on the C^r function space with a metric that takes into account the function (i.e., element of the space C^r) and its first r derivatives. Here the exact structure of the metric in of no significance. See smooth function for the notion of C^r function.

Notice that the C^r topology is in fact a metric space, but in the context of this concept only the topology induced by the metric space is of interest.

It is worthwhile to study the topology because C^r is an infinite separable Banach (complete metric) space, so the norms (metric) are not equivalent, so for each different norm we get different metric space. However, the topology of such spaces are equivalent.