What is complete?

Complete (Mathematics)

In mathematics, "complete" has several distinct meanings, depending on the context. Here are some of the most common:

• <a href="https://www.wikiwhat.page/kavramlar/Complete%20Metric%20Space">Complete Metric Space</a>: A metric space in which every Cauchy sequence converges to a point within the space. Intuitively, this means there are no "missing" points that limit the convergence of sequences. Examples include the real numbers with the usual metric and Euclidean space.

• <a href="https://www.wikiwhat.page/kavramlar/Complete%20Ordered%20Field">Complete Ordered Field</a>: An ordered field where every non-empty subset that is bounded above has a least upper bound (supremum). The real numbers are the only complete ordered field, up to isomorphism.

• <a href="https://www.wikiwhat.page/kavramlar/Complete%20Lattice">Complete Lattice</a>: A partially ordered set in which every subset (not just every finite subset) has both a least upper bound (supremum) and a greatest lower bound (infimum).

• <a href="https://www.wikiwhat.page/kavramlar/Complete%20Graph">Complete Graph</a>: In graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. It's often denoted as K_n, where n is the number of vertices.

• <a href="https://www.wikiwhat.page/kavramlar/Complete%20Measure">Complete Measure</a>: A measure space is complete if every subset of a null set is measurable (and therefore also a null set). Completeness ensures that all "negligible" sets are properly accounted for in the measure.

• <a href="https://www.wikiwhat.page/kavramlar/Complete%20Theory">Complete Theory</a>: In logic, a theory T is complete if, for every sentence φ in the language of T, either φ is provable from T or its negation ¬φ is provable from T. This means the theory answers every question expressible in its language.