Definitions/Role in mathematics/Explanation/Remark

In mathematics, definitions are usually presented as such and get a number so that it is easy to refer to them. The definition contains the description of a situation where only concepts are used that have been defined before. In this situation, a new concept together with a name for it is introduced. This name is printed in a certain font, typically in italic. The new concept can be formulated without the new name; the new name is an abbreviation for the new concept. Quite often, the concepts depend on parameters, like the product set depends on its component sets. The names are often chosen arbitrarily; the meaning of the word within the mathematical context can be understood only via the explicit definition and not via its meaning in everyday life.