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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | imacmp 22901 | The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
β’ ((πΉ β (π½ Cn πΎ) β§ (π½ βΎt π΄) β Comp) β (πΎ βΎt (πΉ β π΄)) β Comp) | ||
Theorem | discmp 22902 | A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ (π΄ β Fin β π« π΄ β Comp) | ||
Theorem | cmpsublem 22903* | Lemma for cmpsub 22904. (Contributed by Jeff Hankins, 28-Jun-2009.) |
β’ π = βͺ π½ β β’ ((π½ β Top β§ π β π) β (βπ β π« π½(π β βͺ π β βπ β (π« π β© Fin)π β βͺ π) β βπ β π« (π½ βΎt π)(βͺ (π½ βΎt π) = βͺ π β βπ‘ β (π« π β© Fin)βͺ (π½ βΎt π) = βͺ π‘))) | ||
Theorem | cmpsub 22904* | Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
β’ π = βͺ π½ β β’ ((π½ β Top β§ π β π) β ((π½ βΎt π) β Comp β βπ β π« π½(π β βͺ π β βπ β (π« π β© Fin)π β βͺ π))) | ||
Theorem | tgcmp 22905* | A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 23549, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ ((π΅ β TopBases β§ π = βͺ π΅) β ((topGenβπ΅) β Comp β βπ¦ β π« π΅(π = βͺ π¦ β βπ§ β (π« π¦ β© Fin)π = βͺ π§))) | ||
Theorem | cmpcld 22906 | A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.) |
β’ ((π½ β Comp β§ π β (Clsdβπ½)) β (π½ βΎt π) β Comp) | ||
Theorem | uncmp 22907 | The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.) |
β’ π = βͺ π½ β β’ (((π½ β Top β§ π = (π βͺ π)) β§ ((π½ βΎt π) β Comp β§ (π½ βΎt π) β Comp)) β π½ β Comp) | ||
Theorem | fiuncmp 22908* | A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ π = βͺ π½ β β’ ((π½ β Top β§ π΄ β Fin β§ βπ₯ β π΄ (π½ βΎt π΅) β Comp) β (π½ βΎt βͺ π₯ β π΄ π΅) β Comp) | ||
Theorem | sscmp 22909 | A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.) |
β’ π = βͺ πΎ β β’ ((π½ β (TopOnβπ) β§ πΎ β Comp β§ π½ β πΎ) β π½ β Comp) | ||
Theorem | hauscmplem 22910* | Lemma for hauscmp 22911. (Contributed by Mario Carneiro, 27-Nov-2013.) |
β’ π = βͺ π½ & β’ π = {π¦ β π½ β£ βπ€ β π½ (π΄ β π€ β§ ((clsβπ½)βπ€) β (π β π¦))} & β’ (π β π½ β Haus) & β’ (π β π β π) & β’ (π β (π½ βΎt π) β Comp) & β’ (π β π΄ β (π β π)) β β’ (π β βπ§ β π½ (π΄ β π§ β§ ((clsβπ½)βπ§) β (π β π))) | ||
Theorem | hauscmp 22911 | A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.) |
β’ π = βͺ π½ β β’ ((π½ β Haus β§ π β π β§ (π½ βΎt π) β Comp) β π β (Clsdβπ½)) | ||
Theorem | cmpfi 22912* | If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
β’ (π½ β Top β (π½ β Comp β βπ₯ β π« (Clsdβπ½)(Β¬ β β (fiβπ₯) β β© π₯ β β ))) | ||
Theorem | cmpfii 22913 | In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
β’ ((π½ β Comp β§ π β (Clsdβπ½) β§ Β¬ β β (fiβπ)) β β© π β β ) | ||
Theorem | bwth 22914* | The glorious Bolzano-Weierstrass theorem. The first general topology theorem ever proved. The first mention of this theorem can be found in a course by Weierstrass from 1865. In his course Weierstrass called it a lemma. He didn't know how famous this theorem would be. He used a Euclidean space instead of a general compact space. And he was not aware of the Heine-Borel property. But the concepts of neighborhood and limit point were already there although not precisely defined. Cantor was one of his students. He published and used the theorem in an article from 1872. The rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) Revised by BL to significantly shorten the proof and avoid infinity, regularity, and choice. (Revised by Brendan Leahy, 26-Dec-2018.) |
β’ π = βͺ π½ β β’ ((π½ β Comp β§ π΄ β π β§ Β¬ π΄ β Fin) β βπ₯ β π π₯ β ((limPtβπ½)βπ΄)) | ||
Syntax | cconn 22915 | Extend class notation with the class of all connected topologies. |
class Conn | ||
Definition | df-conn 22916 | Topologies are connected when only β and βͺ π are both open and closed. (Contributed by FL, 17-Nov-2008.) |
β’ Conn = {π β Top β£ (π β© (Clsdβπ)) = {β , βͺ π}} | ||
Theorem | isconn 22917 | The predicate π½ is a connected topology . (Contributed by FL, 17-Nov-2008.) |
β’ π = βͺ π½ β β’ (π½ β Conn β (π½ β Top β§ (π½ β© (Clsdβπ½)) = {β , π})) | ||
Theorem | isconn2 22918 | The predicate π½ is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.) |
β’ π = βͺ π½ β β’ (π½ β Conn β (π½ β Top β§ (π½ β© (Clsdβπ½)) β {β , π})) | ||
Theorem | connclo 22919 | The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.) |
β’ π = βͺ π½ & β’ (π β π½ β Conn) & β’ (π β π΄ β π½) & β’ (π β π΄ β β ) & β’ (π β π΄ β (Clsdβπ½)) β β’ (π β π΄ = π) | ||
Theorem | conndisj 22920 | If a topology is connected, its underlying set can't be partitioned into two nonempty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
β’ π = βͺ π½ & β’ (π β π½ β Conn) & β’ (π β π΄ β π½) & β’ (π β π΄ β β ) & β’ (π β π΅ β π½) & β’ (π β π΅ β β ) & β’ (π β (π΄ β© π΅) = β ) β β’ (π β (π΄ βͺ π΅) β π) | ||
Theorem | conntop 22921 | A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.) |
β’ (π½ β Conn β π½ β Top) | ||
Theorem | indisconn 22922 | The indiscrete topology (or trivial topology) on any set is connected. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ {β , π΄} β Conn | ||
Theorem | dfconn2 22923* | An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
β’ (π½ β (TopOnβπ) β (π½ β Conn β βπ₯ β π½ βπ¦ β π½ ((π₯ β β β§ π¦ β β β§ (π₯ β© π¦) = β ) β (π₯ βͺ π¦) β π))) | ||
Theorem | connsuba 22924* | Connectedness for a subspace. See connsub 22925. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
β’ ((π½ β (TopOnβπ) β§ π΄ β π) β ((π½ βΎt π΄) β Conn β βπ₯ β π½ βπ¦ β π½ (((π₯ β© π΄) β β β§ (π¦ β© π΄) β β β§ ((π₯ β© π¦) β© π΄) = β ) β ((π₯ βͺ π¦) β© π΄) β π΄))) | ||
Theorem | connsub 22925* | Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
β’ ((π½ β (TopOnβπ) β§ π β π) β ((π½ βΎt π) β Conn β βπ₯ β π½ βπ¦ β π½ (((π₯ β© π) β β β§ (π¦ β© π) β β β§ (π₯ β© π¦) β (π β π)) β Β¬ π β (π₯ βͺ π¦)))) | ||
Theorem | cnconn 22926 | Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
β’ π = βͺ πΎ β β’ ((π½ β Conn β§ πΉ:πβontoβπ β§ πΉ β (π½ Cn πΎ)) β πΎ β Conn) | ||
Theorem | nconnsubb 22927 | Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
β’ (π β π½ β (TopOnβπ)) & β’ (π β π΄ β π) & β’ (π β π β π½) & β’ (π β π β π½) & β’ (π β (π β© π΄) β β ) & β’ (π β (π β© π΄) β β ) & β’ (π β ((π β© π) β© π΄) = β ) & β’ (π β π΄ β (π βͺ π)) β β’ (π β Β¬ (π½ βΎt π΄) β Conn) | ||
Theorem | connsubclo 22928 | If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.) |
β’ π = βͺ π½ & β’ (π β π΄ β π) & β’ (π β (π½ βΎt π΄) β Conn) & β’ (π β π΅ β π½) & β’ (π β (π΅ β© π΄) β β ) & β’ (π β π΅ β (Clsdβπ½)) β β’ (π β π΄ β π΅) | ||
Theorem | connima 22929 | The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
β’ π = βͺ π½ & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β π΄ β π) & β’ (π β (π½ βΎt π΄) β Conn) β β’ (π β (πΎ βΎt (πΉ β π΄)) β Conn) | ||
Theorem | conncn 22930 | A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.) |
β’ π = βͺ π½ & β’ (π β π½ β Conn) & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β π β πΎ) & β’ (π β π β (ClsdβπΎ)) & β’ (π β π΄ β π) & β’ (π β (πΉβπ΄) β π) β β’ (π β πΉ:πβΆπ) | ||
Theorem | iunconnlem 22931* | Lemma for iunconn 22932. (Contributed by Mario Carneiro, 11-Jun-2014.) |
β’ (π β π½ β (TopOnβπ)) & β’ ((π β§ π β π΄) β π΅ β π) & β’ ((π β§ π β π΄) β π β π΅) & β’ ((π β§ π β π΄) β (π½ βΎt π΅) β Conn) & β’ (π β π β π½) & β’ (π β π β π½) & β’ (π β (π β© βͺ π β π΄ π΅) β β ) & β’ (π β (π β© π) β (π β βͺ π β π΄ π΅)) & β’ (π β βͺ π β π΄ π΅ β (π βͺ π)) & β’ β²ππ β β’ (π β Β¬ π β π) | ||
Theorem | iunconn 22932* | The indexed union of connected overlapping subspaces sharing a common point is connected. (Contributed by Mario Carneiro, 11-Jun-2014.) |
β’ (π β π½ β (TopOnβπ)) & β’ ((π β§ π β π΄) β π΅ β π) & β’ ((π β§ π β π΄) β π β π΅) & β’ ((π β§ π β π΄) β (π½ βΎt π΅) β Conn) β β’ (π β (π½ βΎt βͺ π β π΄ π΅) β Conn) | ||
Theorem | unconn 22933 | The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.) |
β’ ((π½ β (TopOnβπ) β§ (π΄ β π β§ π΅ β π) β§ (π΄ β© π΅) β β ) β (((π½ βΎt π΄) β Conn β§ (π½ βΎt π΅) β Conn) β (π½ βΎt (π΄ βͺ π΅)) β Conn)) | ||
Theorem | clsconn 22934 | The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ (π½ βΎt π΄) β Conn) β (π½ βΎt ((clsβπ½)βπ΄)) β Conn) | ||
Theorem | conncompid 22935* | The connected component containing π΄ contains π΄. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ π = βͺ {π₯ β π« π β£ (π΄ β π₯ β§ (π½ βΎt π₯) β Conn)} β β’ ((π½ β (TopOnβπ) β§ π΄ β π) β π΄ β π) | ||
Theorem | conncompconn 22936* | The connected component containing π΄ is connected. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ π = βͺ {π₯ β π« π β£ (π΄ β π₯ β§ (π½ βΎt π₯) β Conn)} β β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π) β Conn) | ||
Theorem | conncompss 22937* | The connected component containing π΄ is a superset of any other connected set containing π΄. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ π = βͺ {π₯ β π« π β£ (π΄ β π₯ β§ (π½ βΎt π₯) β Conn)} β β’ ((π β π β§ π΄ β π β§ (π½ βΎt π) β Conn) β π β π) | ||
Theorem | conncompcld 22938* | The connected component containing π΄ is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.) |
β’ π = βͺ {π₯ β π« π β£ (π΄ β π₯ β§ (π½ βΎt π₯) β Conn)} β β’ ((π½ β (TopOnβπ) β§ π΄ β π) β π β (Clsdβπ½)) | ||
Theorem | conncompclo 22939* | The connected component containing π΄ is a subset of any clopen set containing π΄. (Contributed by Mario Carneiro, 20-Sep-2015.) |
β’ π = βͺ {π₯ β π« π β£ (π΄ β π₯ β§ (π½ βΎt π₯) β Conn)} β β’ ((π½ β (TopOnβπ) β§ π β (π½ β© (Clsdβπ½)) β§ π΄ β π) β π β π) | ||
Theorem | t1connperf 22940 | A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.) |
β’ π = βͺ π½ β β’ ((π½ β Fre β§ π½ β Conn β§ Β¬ π β 1o) β π½ β Perf) | ||
Syntax | c1stc 22941 | Extend class definition to include the class of all first-countable topologies. |
class 1stΟ | ||
Syntax | c2ndc 22942 | Extend class definition to include the class of all second-countable topologies. |
class 2ndΟ | ||
Definition | df-1stc 22943* | Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.) |
β’ 1stΟ = {π β Top β£ βπ₯ β βͺ πβπ¦ β π« π(π¦ βΌ Ο β§ βπ§ β π (π₯ β π§ β π₯ β βͺ (π¦ β© π« π§)))} | ||
Definition | df-2ndc 22944* | Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.) |
β’ 2ndΟ = {π β£ βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π)} | ||
Theorem | is1stc 22945* | The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.) |
β’ π = βͺ π½ β β’ (π½ β 1stΟ β (π½ β Top β§ βπ₯ β π βπ¦ β π« π½(π¦ βΌ Ο β§ βπ§ β π½ (π₯ β π§ β π₯ β βͺ (π¦ β© π« π§))))) | ||
Theorem | is1stc2 22946* | An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
β’ π = βͺ π½ β β’ (π½ β 1stΟ β (π½ β Top β§ βπ₯ β π βπ¦ β π« π½(π¦ βΌ Ο β§ βπ§ β π½ (π₯ β π§ β βπ€ β π¦ (π₯ β π€ β§ π€ β π§))))) | ||
Theorem | 1stctop 22947 | A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
β’ (π½ β 1stΟ β π½ β Top) | ||
Theorem | 1stcclb 22948* | A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
β’ π = βͺ π½ β β’ ((π½ β 1stΟ β§ π΄ β π) β βπ₯ β π« π½(π₯ βΌ Ο β§ βπ¦ β π½ (π΄ β π¦ β βπ§ β π₯ (π΄ β π§ β§ π§ β π¦)))) | ||
Theorem | 1stcfb 22949* | For any point π΄ in a first-countable topology, there is a function π:ββΆπ½ enumerating neighborhoods of π΄ which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.) |
β’ π = βͺ π½ β β’ ((π½ β 1stΟ β§ π΄ β π) β βπ(π:ββΆπ½ β§ βπ β β (π΄ β (πβπ) β§ (πβ(π + 1)) β (πβπ)) β§ βπ¦ β π½ (π΄ β π¦ β βπ β β (πβπ) β π¦))) | ||
Theorem | is2ndc 22950* | The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
β’ (π½ β 2ndΟ β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) | ||
Theorem | 2ndctop 22951 | A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
β’ (π½ β 2ndΟ β π½ β Top) | ||
Theorem | 2ndci 22952 | A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.) |
β’ ((π΅ β TopBases β§ π΅ βΌ Ο) β (topGenβπ΅) β 2ndΟ) | ||
Theorem | 2ndcsb 22953* | Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
β’ (π½ β 2ndΟ β βπ₯(π₯ βΌ Ο β§ (topGenβ(fiβπ₯)) = π½)) | ||
Theorem | 2ndcredom 22954 | A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.) |
β’ (π½ β 2ndΟ β π½ βΌ β) | ||
Theorem | 2ndc1stc 22955 | A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) |
β’ (π½ β 2ndΟ β π½ β 1stΟ) | ||
Theorem | 1stcrestlem 22956* | Lemma for 1stcrest 22957. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ (π΅ βΌ Ο β ran (π₯ β π΅ β¦ πΆ) βΌ Ο) | ||
Theorem | 1stcrest 22957 | A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
β’ ((π½ β 1stΟ β§ π΄ β π) β (π½ βΎt π΄) β 1stΟ) | ||
Theorem | 2ndcrest 22958 | A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
β’ ((π½ β 2ndΟ β§ π΄ β π) β (π½ βΎt π΄) β 2ndΟ) | ||
Theorem | 2ndcctbss 22959* | If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
β’ π½ = (topGenβπ΅) & β’ π = {β¨π’, π£β© β£ (π’ β π β§ π£ β π β§ βπ€ β π΅ (π’ β π€ β§ π€ β π£))} β β’ ((π΅ β TopBases β§ π½ β 2ndΟ) β βπ β TopBases (π βΌ Ο β§ π β π΅ β§ π½ = (topGenβπ))) | ||
Theorem | 2ndcdisj 22960* | Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.) |
β’ ((π½ β 2ndΟ β§ βπ₯ β π΄ π΅ β (π½ β {β }) β§ βπ¦β*π₯ β π΄ π¦ β π΅) β π΄ βΌ Ο) | ||
Theorem | 2ndcdisj2 22961* | Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.) |
β’ ((π½ β 2ndΟ β§ π΄ β π½ β§ βπ¦β*π₯ β π΄ π¦ β π₯) β π΄ βΌ Ο) | ||
Theorem | 2ndcomap 22962* | A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.) |
β’ π = βͺ πΎ & β’ (π β π½ β 2ndΟ) & β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β ran πΉ = π) & β’ ((π β§ π₯ β π½) β (πΉ β π₯) β πΎ) β β’ (π β πΎ β 2ndΟ) | ||
Theorem | 2ndcsep 22963* | A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.) |
β’ π = βͺ π½ β β’ (π½ β 2ndΟ β βπ₯ β π« π(π₯ βΌ Ο β§ ((clsβπ½)βπ₯) = π)) | ||
Theorem | dis2ndc 22964 | A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.) |
β’ (π βΌ Ο β π« π β 2ndΟ) | ||
Theorem | 1stcelcls 22965* | A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 10430. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.) |
β’ π = βͺ π½ β β’ ((π½ β 1stΟ β§ π β π) β (π β ((clsβπ½)βπ) β βπ(π:ββΆπ β§ π(βπ‘βπ½)π))) | ||
Theorem | 1stccnp 22966* | A mapping is continuous at π in a first-countable space π iff it is sequentially continuous at π, meaning that the image under πΉ of every sequence converging at π converges to πΉ(π). This proof uses ax-cc 10430, but only via 1stcelcls 22965, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.) |
β’ (π β π½ β 1stΟ) & β’ (π β π½ β (TopOnβπ)) & β’ (π β πΎ β (TopOnβπ)) & β’ (π β π β π) β β’ (π β (πΉ β ((π½ CnP πΎ)βπ) β (πΉ:πβΆπ β§ βπ((π:ββΆπ β§ π(βπ‘βπ½)π) β (πΉ β π)(βπ‘βπΎ)(πΉβπ))))) | ||
Theorem | 1stccn 22967* | A mapping πβΆπ, where π is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence π(π) converging to π₯, its image under πΉ converges to πΉ(π₯). (Contributed by Mario Carneiro, 7-Sep-2015.) |
β’ (π β π½ β 1stΟ) & β’ (π β π½ β (TopOnβπ)) & β’ (π β πΎ β (TopOnβπ)) & β’ (π β πΉ:πβΆπ) β β’ (π β (πΉ β (π½ Cn πΎ) β βπ(π:ββΆπ β βπ₯(π(βπ‘βπ½)π₯ β (πΉ β π)(βπ‘βπΎ)(πΉβπ₯))))) | ||
Syntax | clly 22968 | Extend class notation with the "locally π΄ " predicate of a topological space. |
class Locally π΄ | ||
Syntax | cnlly 22969 | Extend class notation with the "N-locally π΄ " predicate of a topological space. |
class π-Locally π΄ | ||
Definition | df-lly 22970* | Define a space that is locally π΄, where π΄ is a topological property like "compact", "connected", or "path-connected". A topological space is locally π΄ if every neighborhood of a point contains an open subneighborhood that is π΄ in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ Locally π΄ = {π β Top β£ βπ₯ β π βπ¦ β π₯ βπ’ β (π β© π« π₯)(π¦ β π’ β§ (π βΎt π’) β π΄)} | ||
Definition | df-nlly 22971* |
Define a space that is n-locally π΄, where π΄ is a topological
property like "compact", "connected", or
"path-connected". A
topological space is n-locally π΄ if every neighborhood of a point
contains a subneighborhood that is π΄ in the subspace topology.
The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally π΄". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually π-Locally Comp in our terminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ π-Locally π΄ = {π β Top β£ βπ₯ β π βπ¦ β π₯ βπ’ β (((neiβπ)β{π¦}) β© π« π₯)(π βΎt π’) β π΄} | ||
Theorem | islly 22972* | The property of being a locally π΄ topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β Locally π΄ β (π½ β Top β§ βπ₯ β π½ βπ¦ β π₯ βπ’ β (π½ β© π« π₯)(π¦ β π’ β§ (π½ βΎt π’) β π΄))) | ||
Theorem | isnlly 22973* | The property of being an n-locally π΄ topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β π-Locally π΄ β (π½ β Top β§ βπ₯ β π½ βπ¦ β π₯ βπ’ β (((neiβπ½)β{π¦}) β© π« π₯)(π½ βΎt π’) β π΄)) | ||
Theorem | llyeq 22974 | Equality theorem for the Locally π΄ predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π΄ = π΅ β Locally π΄ = Locally π΅) | ||
Theorem | nllyeq 22975 | Equality theorem for the Locally π΄ predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π΄ = π΅ β π-Locally π΄ = π-Locally π΅) | ||
Theorem | llytop 22976 | A locally π΄ space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β Locally π΄ β π½ β Top) | ||
Theorem | nllytop 22977 | A locally π΄ space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β π-Locally π΄ β π½ β Top) | ||
Theorem | llyi 22978* | The property of a locally π΄ topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β Locally π΄ β§ π β π½ β§ π β π) β βπ’ β π½ (π’ β π β§ π β π’ β§ (π½ βΎt π’) β π΄)) | ||
Theorem | nllyi 22979* | The property of an n-locally π΄ topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β π-Locally π΄ β§ π β π½ β§ π β π) β βπ’ β ((neiβπ½)β{π})(π’ β π β§ (π½ βΎt π’) β π΄)) | ||
Theorem | nlly2i 22980* | Eliminate the neighborhood symbol from nllyi 22979. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β π-Locally π΄ β§ π β π½ β§ π β π) β βπ β π« πβπ’ β π½ (π β π’ β§ π’ β π β§ (π½ βΎt π ) β π΄)) | ||
Theorem | llynlly 22981 | A locally π΄ space is n-locally π΄: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β Locally π΄ β π½ β π-Locally π΄) | ||
Theorem | llyssnlly 22982 | A locally π΄ space is n-locally π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ Locally π΄ β π-Locally π΄ | ||
Theorem | llyss 22983 | The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π΄ β π΅ β Locally π΄ β Locally π΅) | ||
Theorem | nllyss 22984 | The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π΄ β π΅ β π-Locally π΄ β π-Locally π΅) | ||
Theorem | subislly 22985* | The property of a subspace being locally π΄. (Contributed by Mario Carneiro, 10-Mar-2015.) |
β’ ((π½ β Top β§ π΅ β π) β ((π½ βΎt π΅) β Locally π΄ β βπ₯ β π½ βπ¦ β (π₯ β© π΅)βπ’ β π½ ((π’ β© π΅) β π₯ β§ π¦ β π’ β§ (π½ βΎt (π’ β© π΅)) β π΄))) | ||
Theorem | restnlly 22986* | If the property π΄ passes to open subspaces, then a space is n-locally π΄ iff it is locally π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π β§ (π β π΄ β§ π₯ β π)) β (π βΎt π₯) β π΄) β β’ (π β π-Locally π΄ = Locally π΄) | ||
Theorem | restlly 22987* | If the property π΄ passes to open subspaces, then a space which is π΄ is also locally π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π β§ (π β π΄ β§ π₯ β π)) β (π βΎt π₯) β π΄) & β’ (π β π΄ β Top) β β’ (π β π΄ β Locally π΄) | ||
Theorem | islly2 22988* | An alternative expression for π½ β Locally π΄ when π΄ passes to open subspaces: A space is locally π΄ if every point is contained in an open neighborhood with property π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π β§ (π β π΄ β§ π₯ β π)) β (π βΎt π₯) β π΄) & β’ π = βͺ π½ β β’ (π β (π½ β Locally π΄ β (π½ β Top β§ βπ¦ β π βπ’ β π½ (π¦ β π’ β§ (π½ βΎt π’) β π΄)))) | ||
Theorem | llyrest 22989 | An open subspace of a locally π΄ space is also locally π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β Locally π΄ β§ π΅ β π½) β (π½ βΎt π΅) β Locally π΄) | ||
Theorem | nllyrest 22990 | An open subspace of an n-locally π΄ space is also n-locally π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β π-Locally π΄ β§ π΅ β π½) β (π½ βΎt π΅) β π-Locally π΄) | ||
Theorem | loclly 22991 | If π΄ is a local property, then both Locally π΄ and π-Locally π΄ simplify to π΄. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (Locally π΄ = π΄ β π-Locally π΄ = π΄) | ||
Theorem | llyidm 22992 | Idempotence of the "locally" predicate, i.e. being "locally π΄ " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ Locally Locally π΄ = Locally π΄ | ||
Theorem | nllyidm 22993 | Idempotence of the "n-locally" predicate, i.e. being "n-locally π΄ " is a local property. (Use loclly 22991 to show π-Locally π-Locally π΄ = π-Locally π΄.) (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ Locally π-Locally π΄ = π-Locally π΄ | ||
Theorem | toplly 22994 | A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ Locally Top = Top | ||
Theorem | topnlly 22995 | A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ π-Locally Top = Top | ||
Theorem | hauslly 22996 | A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β Haus β π½ β Locally Haus) | ||
Theorem | hausnlly 22997 | A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ (π½ β π-Locally Haus β π½ β Locally Haus) | ||
Theorem | hausllycmp 22998 | A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β Haus β§ π½ β Comp) β π½ β π-Locally Comp) | ||
Theorem | cldllycmp 22999 | A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 22990.) (Contributed by Mario Carneiro, 2-Mar-2015.) |
β’ ((π½ β π-Locally Comp β§ π΄ β (Clsdβπ½)) β (π½ βΎt π΄) β π-Locally Comp) | ||
Theorem | lly1stc 23000 | First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.) |
β’ Locally 1stΟ = 1stΟ |
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