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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisfne4 26201 The predicate " is finer than " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremisfne4b 26202 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremisfne2 26203* The predicate " is finer than ." (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremisfne3 26204* The predicate " is finer than ." (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremfnebas 26205 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)

Theoremfnetg 26206 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfnessex 26207* If is finer than and is an element of , every point in is an element of a subset of which is in . (Contributed by Jeff Hankins, 28-Sep-2009.)

Theoremfneuni 26208* If is finer than , every element of is a union of elements of . (Contributed by Jeff Hankins, 11-Oct-2009.)

Theoremfneint 26209* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)

Theoremrefrel 26210 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremisref 26211* The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 26200. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremrefbas 26212 A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremrefssex 26213* Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremfness 26214 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)

Theoremssref 26215 A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremfneref 26216 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)

Theoremrefref 26217 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremfnetr 26218 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremfneval 26219 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfneer 26220 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremreftr 26221 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremtopfne 26222 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)

Theoremtopfneec 26223 A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremtopfneec2 26224 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)

Theoremfnessref 26225* A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremrefssfne 26226* A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)

Theoremisptfin 26227* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremislocfin 26228* The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremfinptfin 26229 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremptfinfin 26230* A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremfinlocfin 26231 A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremlocfintop 26232 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremlocfinbas 26233 A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremlocfinnei 26234* A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Theoremlfinpfin 26235 A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremlocfincmp 26236 For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremlocfindis 26237 The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremlocfincf 26238 A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
TopOn

Theoremcomppfsc 26239* A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

19.13.5  Neighborhood bases determine topologies

Theoremneibastop1 26240* A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
TopOn

Theoremneibastop2lem 26241* Lemma for neibastop2 26242. (Contributed by Jeff Hankins, 12-Sep-2009.)

Theoremneibastop2 26242* In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Theoremneibastop3 26243* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
TopOn

19.13.6  Lattice structure of topologies

Theoremtopmtcl 26244 The meet of a collection of topologies on is again a topology on . (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
TopOn TopOn

Theoremtopmeet 26245* Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
TopOn TopOn

Theoremtopjoin 26246* Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
TopOn TopOn

Theoremfnemeet1 26247* The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremfnemeet2 26248* The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremfnejoin1 26249* Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremfnejoin2 26250* Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

19.13.7  Filter bases

Theoremfgmin 26251 Minimality property of a generated filter: every filter that contains contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)

Theoremneifg 26252* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17813. (Contributed by Jeff Hankins, 3-Sep-2009.)

19.13.8  Directed sets, nets

Theoremtailfval 26253* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtailval 26254 The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremeltail 26255 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtailf 26256 The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtailini 26257 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)

Theoremtailfb 26258 The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem1 26259* Lemma for filnet 26263. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem2 26260* Lemma for filnet 26263. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem3 26261* Lemma for filnet 26263. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnetlem4 26262* Lemma for filnet 26263. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremfilnet 26263* A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

19.14.1  Logic and set theory

Theoremanim12da 26264 Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremsyldanl 26265 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremunirep 26266* Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremcover2 26267* Two ways of expressing the statement "there is a cover of by elements of such that for each set in the cover, ." Note that and must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremcover2g 26268* Two ways of expressing the statement "there is a cover of by elements of such that for each set in the cover, ." Note that and must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)

Theorembrabg2 26269* Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopelopab3 26270* Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)

Theoremcocanfo 26271 Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theorembrresi 26272 Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfnopabeqd 26273* Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremfvopabf4g 26274* Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremeqfnun 26275 Two functions on are equal if and only if they have equal restrictions to both and . (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremfnopabco 26276* Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremopropabco 26277* Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremf1opr 26278* Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremcocnv 26279 Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1ocan1fv 26280 Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremf1ocan2fv 26281 Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminixp 26282* Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremupixp 26283* Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremabrexdom 26284* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabrexdom2 26285* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremac6gf 26286* Axiom of Choice. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremindexa 26287* If for every element of an indexing set there exists a corresponding element of another set , then there exists a subset of consisting only of those elements which are indexed by . Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremindexdom 26288* If for every element of an indexing set there exists a corresponding element of another set , then there exists a subset of consisting only of those elements which are indexed by , and which is dominated by the set . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfrinfm 26289* A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremwelb 26290* A non-empty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupex2g 26291 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupclt 26292* Closure of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupubt 26293* Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

19.14.2  Real and complex numbers; integers

Theoremfilbcmb 26294* Combine a finite set of lower bounds. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremrdr 26295 Two ways of expressing the remainder when is divided by . (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfzmul 26296 Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfzadd2 26297 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

19.14.3  Sequences and sums

Theoremsdclem2 26298* Lemma for sdc 26300. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsdclem1 26299* Lemma for sdc 26300. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsdc 26300* Strong dependent choice. Suppose we may choose an element of such that property holds, and suppose that if we have already chosen the first elements (represented here by a function from to ), we may choose another element so that all elements taken together have property . Then there exists an infinite sequence of elements of such that the first terms of this sequence satisfy for all . This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)

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