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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremishil2 16901* The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremisobs 16902* The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
Scalar                                   OBasis

Theoremobsip 16903 The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
Scalar                     OBasis

Theoremobsipid 16904 A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
Scalar              OBasis

Theoremobsrcl 16905 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobsss 16906 An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobsne0 16907 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobsocv 16908 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobs2ocv 16909 The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobselocv 16910 A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobs2ss 16911 A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis OBasis

Theoremobslbs 16912 An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
LBasis                     OBasis

PART 11  BASIC TOPOLOGY

11.1  Topology

11.1.1  Topological spaces

Syntaxctop 16913 Extend class notation with the class of all topologies.

Syntaxctopon 16914 The class function of all topologies over a base set.
TopOn

SyntaxctpsOLD 16915 Extend class notation with the class of all topological spaces. (New usage is discouraged.)

Syntaxctps 16916 Extend class notation with the class of all topological spaces.

Syntaxctb 16917 Extend class notation with the class of all topological bases.

Definitiondf-top 16918* Define the (proper) class of all topologies. See istop2g 16924 for an alternate way to express finite intersection and istps5OLD 16944 for a standard definition in terms of both members of a topological space. (Contributed by NM, 3-Mar-2006.)

Definitiondf-topspOLD 16919* Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5OLD 16944 for a standard way to express a topological space. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)

Definitiondf-bases 16920* Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 16968). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)

Definitiondf-topon 16921* Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn

Definitiondf-topsp 16922 Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopOn

Theoremistopg 16923* Express the predicate " is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion may have led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremistop2g 16924* Express the predicate " is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)

Theoremuniopn 16925 The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)

Theoremiunopn 16926* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

Theoreminopn 16927 The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)

Theoremfitop 16928 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)

Theoremfiinopn 16929 The intersection of a non-empty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

Theoremiinopn 16930* The intersection of a non-empty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremunopn 16931 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem0opn 16932 The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)

Theorem0ntop 16933 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

Theoremtopopn 16934 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

Theoremeltopss 16935 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)

Theoremriinopn 16936* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremrintopn 16937 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

TheoremeltopspOLD 16938 Construct a topological space from a topology and vice-versa. We say that is a topology on . (This could be proved more efficiently from istpsOLD 16940, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremtpsexOLD 16939 Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremistpsOLD 16940 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps2OLD 16941 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps3OLD 16942* A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps4OLD 16943* A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps5OLD 16944* A standard textbook definition of a topological space : a topology on is a collection of subsets of such that and are in and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistopon 16945 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopontop 16946 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponuni 16947 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponmax 16948 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponss 16949 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtoponcom 16950 If is a topology on the base set of topology , then is a topology on the base of . (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtopontopi 16951 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponunii 16952 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoptopon 16953 Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopgele 16954 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theoremtopsn 16955 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 3969). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theoremistps 16956 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremistps2 16957 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)

Theoremtpsuni 16958 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)

Theoremtpstop 16959 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)

Theoremtpspropd 16960 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtpsprop2d 16961 A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet TopSet

Theoremtopontopn 16962 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet       TopOn

Theoremtsettps 16963 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet       TopOn

Theoremistpsi 16964 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)

Theoremeltpsg 16965 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet        TopOn

Theoremeltpsi 16966 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopSet

11.1.2  TopBases for topologies

Theoremisbasisg 16967* Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)

Theoremisbasis2g 16968* Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)

Theoremisbasis3g 16969* Express the predicate " is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)

Theorembasis1 16970 Property of a basis. (Contributed by NM, 16-Jul-2006.)

Theorembasis2 16971* Property of a basis. (Contributed by NM, 17-Jul-2006.)

Theoremfiinbas 16972* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorembasdif0 16973 A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theorembaspartn 16974* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremtgval 16975* The topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremtgval2 16976* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 16989) that is indeed a topology (on ; see unitg 16987). (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg 16977 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg2 16978* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg2b 16979* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg4i 16980 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremeltg3i 16981 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremeltg3 16982* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)

Theoremtgval3 16983* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtg1 16984 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)

Theoremtg2 16985* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)

Theorembastg 16986 A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremunitg 16987 The topology generated by a basis is a topology on . Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.)

Theoremtgss 16988 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)

Theoremtgcl 16989 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)

Theoremtgclb 16990 The property tgcl 16989 can be reversed: if the topology generated by is actually a topology, then must be a topological basis. This yields an alternative definition of . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtgtopon 16991 A basis generates a topology on . (Contributed by Mario Carneiro, 14-Aug-2015.)
TopOn

Theoremtopbas 16992 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)

Theoremtgtop 16993 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)

Theoremeltop 16994 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)

Theoremeltop2 16995* Membership in a topology. (Contributed by NM, 19-Jul-2006.)

Theoremeltop3 16996* Membership in a topology. (Contributed by NM, 19-Jul-2006.)

Theoremfibas 16997 A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtgdom 16998 A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)

Theoremtgiun 16999* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtgidm 17000 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

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