Equivalent definitions of mathematical structures

In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>), a notion (for example, <a href="/facts/Ellipse/1wUDnGxY">ellipse</a> or <a href="/facts/Minimal_surface/Fy7VkrLQ">minimal surface</a>) may have more than one definition. These definitions are equivalent in the context of a given <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structure</a> (<a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, in this case). Second, a mathematical structure may have more than one definition (for example, <a href="/facts/Topological_space/v2ut7CbK">topological space</a> has at least <a href="/facts/Characterizations_of_the_category_of_topological_spaces/moIyGjVd">seven definitions</a>; <a href="/facts/Ordered_field/an0prSgG">ordered field</a> has at least <a href="/facts/Ordered_field/an0prSgG">two definitions</a>).
In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it satisfies the other definition.
In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object. Many different objects may implement the same structure.

Equivalent definitions of mathematical structures open-in-new

Equivalent definitions of mathematical structures