Revision #688 → #1099 · back to history
addedComplete metric space1e57f9e51a70
addedRationals are not complete (intuitive)779c8ab0626d
addedCauchy sequence in a metric spacec52a2f17aa3e
addedComplete space (equivalent conditions)c9b0cc637df5
addedComplete space — Cauchy sequence conditionfd5ae84cfab3
addedComplete space — nested closed sets condition (Cantor's intersection theorem)048878b76120
addedRationals are not complete90c95cf2b9db
addedOpen interval (0,1) is not completea5521991fd51
addedClosed interval [0,1] is complete2b4576241911
addedReal numbers and complex numbers are completeb4f67f28fe3a
addedEuclidean space is completebc54e14d1914
addedBanach space (complete normed vector space)5e1e49116aff
addedC([a,b]) is a Banach space with supremum norm519d5e154ae4
addedSupremum norm does not give a norm on C(R)538ee61ea566
addedFrechet space14d47e851c6c
addedC(R) is a Frechet space with compact convergence topology591465a045c6
addedp-adic numbers are complete4c7138d758fb
addedSequence space over arbitrary set is completeee4f549aa721
addedGeodesic manifold (complete Riemannian manifold)310fa64b55a2
addedCompleteness of Riemannian manifolds follows from Hopf-Rinowa94d6db51fba
addedEvery compact metric space is completebf02b39fcf19
addedA metric space is compact iff complete and totally boundede7e00f232588
addedHeine-Borel theorem: closed and bounded subspace of R^n is compact1928899d5c26
addedClosed subset of a complete space is completef95fbe64dded
addedComplete subspace of a metric space is closed08699e137799
addedSubspace of complete space is complete iff closed (theorem box)cdc85428402a
addedBounded functions into complete space form a complete space3374fb3c9c19
addedContinuous bounded functions into complete space form closed subspace182a48201725
addedBaire category theorem: complete metric space is a Baire space36ca3415f4fa
addedBanach fixed-point theorem: contraction mapping admits a fixed pointdc727d4a0270
addedFixed-point theorem used to prove inverse function theorembdfcf5461d37
addedUrsescu's theorem (theorem box)860f54414f33
addedExistence of completion for any metric spaced96110ad9c75
addedUniversal property of completion380d5cdcb783
addedCompletion is unique up to isometry90c30cfa64ff
addedCompletion constructed via equivalence classes of Cauchy sequencesa5c0501de62f
addedDistance on Cauchy sequences is a pseudometric40d9c63216ab
addedEmbedding into completion defines an isometry onto dense subspaced5ac04c39b7a
addedReal numbers are completion of rationals0012d6fba3ef
addedReal numbers are the unique totally ordered complete field0c8ef7894947
addedp-adic numbers arise from completing rationals with different metric39e0c82d5cb0
addedCompletion of normed vector space is a Banach spacec87b52276568
addedCompletion of inner product space is a Hilbert space5d1c4d78d537
addedCompleteness is a metric property, not topological9de5290fa986
addedReals are homeomorphic to (0,1) but complete vs not50cb65c20116
addedCompletely metrizable space8c5646262f9f
addedCharacterization of completely metrizable spaces as G-delta setsb1482d98df2e
addedBaire category theorem applies to completely metrizable spacesf25bae7ff3db
addedCompletely uniformizable space67f7a7fab1cc
addedPolish space3610903ea45e
addedCompleteness via group structure (topological groups)6e65297bea14
addedCompleteness in uniform spaces2326750396e0
addedCompleteness via Cauchy nets or Cauchy filters6dcf6c58bc17
addedComplete space via Cauchy nets/filters0ba8d7704127
addedCompletion exists for arbitrary uniform spaces065400d8d701
addedCauchy spaces as most general setting for Cauchy nets94650ff887f2