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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqtopeu 17701* Universal property of the quotient topology. If is a function from to which is equal on all equivalent elements under , then there is a unique continuous map such that , and we say that "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremqtoprest 17702 If is a saturated open or closed set (where saturated means that for some ), then the restriction of the quotient map to is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn                                   qTop t t qTop

Theoremqtopomap 17703* If is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremqtopcmap 17704* If is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremimastopn 17705 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
s                                           qTop

Theoremimastps 17706 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremdivstps 17707 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremkqfval 17708* Value of the function appearing in df-kq 17679. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqfeq 17709* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqffn 17710* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqval 17711* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ qTop

Theoremkqtopon 17712* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ TopOn

Theoremkqid 17713* The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremist0-4 17714* The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqfvima 17715* When the image set is open, the quotient map satisfies a partial converse to fnfvima 5935, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqsat 17716* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17702). (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqdisj 17717* A version of imain 5488 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqcldsat 17718* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17702). (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqopn 17719* The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqcld 17720* The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqt0lem 17721* Lemma for kqt0 17731. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn KQ

Theoremisr0 17722* The property " is an R0 space". A space is R0 if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains also contains , so there is no separation, then and are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R0 if and only if its Kolmogorov quotient is T1, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremr0cld 17723* The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremregr1lem 17724* Lemma for regr1 17735. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn                                   KQ KQ

Theoremregr1lem2 17725* A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqreglem1 17726* A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqreglem2 17727* If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqnrmlem1 17728* A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqnrmlem2 17729* If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqtop 17730 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqt0 17731 The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqf 17732 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremr0sep 17733* The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremnrmr0reg 17734 A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremregr1 17735 A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqreg 17736 The Kolmogorov quotient of a regular space is regular. By regr1 17735 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqnrm 17737 The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

11.1.19  Homeomorphisms

Syntaxchmeo 17738 Extend class notation with the class of all homeomorphisms.

Syntaxchmph 17739 Extend class notation with the relation "is homeomorph to.".

Definitiondf-hmeo 17740* Function returning all the homeomorphisms from topology to topology . (Contributed by FL, 14-Feb-2007.)

Definitiondf-hmph 17741 Definition of the relation is homeomorph to . (Contributed by FL, 14-Feb-2007.)

Theoremhmeofn 17742 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmeofval 17743* The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremishmeo 17744 The predicate F is a homeomorphism between topology and topology . Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocn 17745 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnvcn 17746 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnv 17747 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeof1o2 17748 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremhmeof1o 17749 A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremhmeoima 17750 The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeoopn 17751 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocld 17752 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocls 17753 Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeontr 17754 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeoimaf1o 17755* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmeores 17756 The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
t t

Theoremhmeoco 17757 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremidhmeo 17758 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremhmeocnvb 17759 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmeoqtop 17760 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
qTop

Theoremhmph 17761 Express the predicate is homeomorph to . (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmphi 17762 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmphtop 17763 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmphtop1 17764 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphtop2 17765 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphref 17766 "Is homeomorph to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremhmphsym 17767 "Is homeomorph to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremhmphtr 17768 "Is homeomorph to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmpher 17769 "Is homeomorph to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphen 17770 Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremhmphsymb 17771 "Is homeomorph to" is symmetric. (Contributed by FL, 22-Feb-2007.)

Theoremhaushmphlem 17772* Lemma for haushmph 17777 and similar theorems. If the topological property is preserved under injective preimages, then property is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcmphmph 17773 Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremconhmph 17774 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)

Theoremt0hmph 17775 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremt1hmph 17776 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhaushmph 17777 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremreghmph 17778 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremnrmhmph 17779 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmph0 17780 A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremhmphdis 17781 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmphindis 17782 Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremindishmph 17783 Equinumerous sets equipped with their indiscrete topologies are homeomorph (which means in that particular case that a segment is homeomorph to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

Theoremhmphen2 17784 Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremcmphaushmeo 17785 A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremordthmeolem 17786 Lemma for ordthmeo 17787. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTop

Theoremordthmeo 17787 An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTop

Theoremtxhmeo 17788* Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxswaphmeolem 17789* Show inverse for the "swap components" operation on a cross product. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremtxswaphmeo 17790* There is a homeomorphism from to . (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn TopOn

Theorempt1hmeo 17791* The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.)
TopOn

Theoremptuncnv 17792* Exhibit the converse function of the map which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremptunhmeo 17793* Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of . (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremxpstopnlem1 17794* The function used in xpsval 13752 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
TopOn       TopOn

Theoremxpstps 17795 A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremxpstopnlem2 17796* Lemma for xpstopn 17797. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremxpstopn 17797 The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on to by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremptcmpfi 17798 A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremxkocnv 17799* The inverse of the "currying" function is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
TopOn       TopOn              𝑛Locally        𝑛Locally

Theoremxkohmeo 17800* The Exponential Law for topological spaces. The "currying" function is a homeomorphism on function spaces when and are exponentiable spaces (by xkococn 17645, it is sufficient to assume that are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn       TopOn              𝑛Locally        𝑛Locally

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