Theorem lecldbas 19847

Description: The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
lecldbas.1	|- F = ( x e. ran [,] |-> ( RR* \ x ) )

Assertion

Ref	Expression
lecldbas	|- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Proof of Theorem lecldbas

Dummy variables a b c y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2457	. . . 4 |- ran ( y e. RR* |-> ( y (,] +oo ) ) = ran ( y e. RR* |-> ( y (,] +oo ) )
2		eqid 2457	. . . 4 |- ran ( y e. RR* |-> ( -oo [,) y ) ) = ran ( y e. RR* |-> ( -oo [,) y ) )
3	1, 2	leordtval2 19840	. . 3 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) )
4		fvex 5882	. . . 4 |- ( fi ` ran F ) e. _V
5		fvex 5882	. . . . . 6 |- (ordTop ` <_ ) e. _V
6		lecldbas.1	. . . . . . . 8 |- F = ( x e. ran [,] |-> ( RR* \ x ) )
7		iccf 11648	. . . . . . . . . . 11 |- [,] : ( RR* X. RR* ) --> ~P RR*
8		ffn 5737	. . . . . . . . . . 11 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> [,] Fn ( RR* X. RR* ) )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- [,] Fn ( RR* X. RR* )
10		ovelrn 6450	. . . . . . . . . 10 |- ( [,] Fn ( RR* X. RR* ) -> ( x e. ran [,] <-> E. a e. RR* E. b e. RR* x = ( a [,] b ) ) )
11	9, 10	ax-mp 5	. . . . . . . . 9 |- ( x e. ran [,] <-> E. a e. RR* E. b e. RR* x = ( a [,] b ) )
12		difeq2 3612	. . . . . . . . . . . 12 |- ( x = ( a [,] b ) -> ( RR* \ x ) = ( RR* \ ( a [,] b ) ) )
13		iccordt 19842	. . . . . . . . . . . . 13 |- ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) )
14		letopuni 19835	. . . . . . . . . . . . . 14 |- RR* = U. (ordTop ` <_ )
15	14	cldopn 19659	. . . . . . . . . . . . 13 |- ( ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) ) -> ( RR* \ ( a [,] b ) ) e. (ordTop ` <_ ) )
16	13, 15	ax-mp 5	. . . . . . . . . . . 12 |- ( RR* \ ( a [,] b ) ) e. (ordTop ` <_ )
17	12, 16	syl6eqel 2553	. . . . . . . . . . 11 |- ( x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
18	17	rexlimivw 2946	. . . . . . . . . 10 |- ( E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
19	18	rexlimivw 2946	. . . . . . . . 9 |- ( E. a e. RR* E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
20	11, 19	sylbi 195	. . . . . . . 8 |- ( x e. ran [,] -> ( RR* \ x ) e. (ordTop ` <_ ) )
21	6, 20	fmpti 6055	. . . . . . 7 |- F : ran [,] --> (ordTop ` <_ )
22		frn 5743	. . . . . . 7 |- ( F : ran [,] --> (ordTop ` <_ ) -> ran F C_ (ordTop ` <_ ) )
23	21, 22	ax-mp 5	. . . . . 6 |- ran F C_ (ordTop ` <_ )
24	5, 23	ssexi 4601	. . . . 5 |- ran F e. _V
25		eqid 2457	. . . . . . . 8 |- ( y e. RR* |-> ( y (,] +oo ) ) = ( y e. RR* |-> ( y (,] +oo ) )
26		mnfxr 11348	. . . . . . . . . . 11 |- -oo e. RR*
27		fnovrn 6449	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ -oo e. RR* /\ y e. RR* ) -> ( -oo [,] y ) e. ran [,] )
28	9, 26, 27	mp3an12 1314	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,] y ) e. ran [,] )
29	26	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo e. RR* )
30		id 22	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y e. RR* )
31		pnfxr 11346	. . . . . . . . . . . . . . 15 |- +oo e. RR*
32	31	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> +oo e. RR* )
33		mnfle 11367	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo <_ y )
34		pnfge 11364	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y <_ +oo )
35		df-icc 11561	. . . . . . . . . . . . . . 15 |- [,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c <_ b ) } )
36		df-ioc 11559	. . . . . . . . . . . . . . 15 |- (,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a < c /\ c <_ b ) } )
37		xrltnle 9670	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y < z <-> -. z <_ y ) )
38		xrletr 11386	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z <_ y /\ y <_ +oo ) -> z <_ +oo ) )
39		xrlelttr 11384	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo < z ) )
40		xrltle 11380	. . . . . . . . . . . . . . . . 17 |- ( ( -oo e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
41	40	3adant2 1015	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
42	39, 41	syld 44	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo <_ z ) )
43	35, 36, 37, 35, 38, 42	ixxun 11570	. . . . . . . . . . . . . 14 |- ( ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) /\ ( -oo <_ y /\ y <_ +oo ) ) -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = ( -oo [,] +oo ) )
44	29, 30, 32, 33, 34, 43	syl32anc 1236	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = ( -oo [,] +oo ) )
45		iccmax 11625	. . . . . . . . . . . . 13 |- ( -oo [,] +oo ) = RR*
46	44, 45	syl6eq 2514	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* )
47		iccssxr 11632	. . . . . . . . . . . . 13 |- ( -oo [,] y ) C_ RR*
48	35, 36, 37	ixxdisj 11569	. . . . . . . . . . . . . 14 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
49	26, 31, 48	mp3an13 1315	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
50		uneqdifeq 3919	. . . . . . . . . . . . 13 |- ( ( ( -oo [,] y ) C_ RR* /\ ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) ) -> ( ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* <-> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) ) )
51	47, 49, 50	sylancr 663	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* <-> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) ) )
52	46, 51	mpbid 210	. . . . . . . . . . 11 |- ( y e. RR* -> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) )
53	52	eqcomd 2465	. . . . . . . . . 10 |- ( y e. RR* -> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) )
54		difeq2 3612	. . . . . . . . . . . 12 |- ( x = ( -oo [,] y ) -> ( RR* \ x ) = ( RR* \ ( -oo [,] y ) ) )
55	54	eqeq2d 2471	. . . . . . . . . . 11 |- ( x = ( -oo [,] y ) -> ( ( y (,] +oo ) = ( RR* \ x ) <-> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) )
56	55	rspcev 3210	. . . . . . . . . 10 |- ( ( ( -oo [,] y ) e. ran [,] /\ ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
57	28, 53, 56	syl2anc 661	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
58		xrex 11242	. . . . . . . . . . 11 |- RR* e. _V
59		difexg 4604	. . . . . . . . . . 11 |- ( RR* e. _V -> ( RR* \ x ) e. _V )
60	58, 59	ax-mp 5	. . . . . . . . . 10 |- ( RR* \ x ) e. _V
61	6, 60	elrnmpti 5263	. . . . . . . . 9 |- ( ( y (,] +oo ) e. ran F <-> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
62	57, 61	sylibr 212	. . . . . . . 8 |- ( y e. RR* -> ( y (,] +oo ) e. ran F )
63	25, 62	fmpti 6055	. . . . . . 7 |- ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F
64		frn 5743	. . . . . . 7 |- ( ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F -> ran ( y e. RR* |-> ( y (,] +oo ) ) C_ ran F )
65	63, 64	ax-mp 5	. . . . . 6 |- ran ( y e. RR* |-> ( y (,] +oo ) ) C_ ran F
66		eqid 2457	. . . . . . . 8 |- ( y e. RR* |-> ( -oo [,) y ) ) = ( y e. RR* |-> ( -oo [,) y ) )
67		fnovrn 6449	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ y e. RR* /\ +oo e. RR* ) -> ( y [,] +oo ) e. ran [,] )
68	9, 31, 67	mp3an13 1315	. . . . . . . . . 10 |- ( y e. RR* -> ( y [,] +oo ) e. ran [,] )
69		df-ico 11560	. . . . . . . . . . . . . . 15 |- [,) = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c < b ) } )
70		xrlenlt 9669	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y <_ z <-> -. z < y ) )
71		xrltletr 11385	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z < +oo ) )
72		xrltle 11380	. . . . . . . . . . . . . . . . 17 |- ( ( z e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
73	72	3adant2 1015	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
74	71, 73	syld 44	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z <_ +oo ) )
75		xrletr 11386	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y <_ z ) -> -oo <_ z ) )
76	69, 35, 70, 35, 74, 75	ixxun 11570	. . . . . . . . . . . . . 14 |- ( ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) /\ ( -oo <_ y /\ y <_ +oo ) ) -> ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( -oo [,] +oo ) )
77	29, 30, 32, 33, 34, 76	syl32anc 1236	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( -oo [,] +oo ) )
78		uncom 3644	. . . . . . . . . . . . 13 |- ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( ( y [,] +oo ) u. ( -oo [,) y ) )
79	77, 78, 45	3eqtr3g 2521	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* )
80		iccssxr 11632	. . . . . . . . . . . . 13 |- ( y [,] +oo ) C_ RR*
81		incom 3687	. . . . . . . . . . . . . 14 |- ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = ( ( -oo [,) y ) i^i ( y [,] +oo ) )
82	69, 35, 70	ixxdisj 11569	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
83	26, 31, 82	mp3an13 1315	. . . . . . . . . . . . . 14 |- ( y e. RR* -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
84	81, 83	syl5eq 2510	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) )
85		uneqdifeq 3919	. . . . . . . . . . . . 13 |- ( ( ( y [,] +oo ) C_ RR* /\ ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) ) -> ( ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* <-> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) ) )
86	80, 84, 85	sylancr 663	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* <-> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) ) )
87	79, 86	mpbid 210	. . . . . . . . . . 11 |- ( y e. RR* -> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) )
88	87	eqcomd 2465	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) )
89		difeq2 3612	. . . . . . . . . . . 12 |- ( x = ( y [,] +oo ) -> ( RR* \ x ) = ( RR* \ ( y [,] +oo ) ) )
90	89	eqeq2d 2471	. . . . . . . . . . 11 |- ( x = ( y [,] +oo ) -> ( ( -oo [,) y ) = ( RR* \ x ) <-> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) )
91	90	rspcev 3210	. . . . . . . . . 10 |- ( ( ( y [,] +oo ) e. ran [,] /\ ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
92	68, 88, 91	syl2anc 661	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
93	6, 60	elrnmpti 5263	. . . . . . . . 9 |- ( ( -oo [,) y ) e. ran F <-> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
94	92, 93	sylibr 212	. . . . . . . 8 |- ( y e. RR* -> ( -oo [,) y ) e. ran F )
95	66, 94	fmpti 6055	. . . . . . 7 |- ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F
96		frn 5743	. . . . . . 7 |- ( ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F -> ran ( y e. RR* |-> ( -oo [,) y ) ) C_ ran F )
97	95, 96	ax-mp 5	. . . . . 6 |- ran ( y e. RR* |-> ( -oo [,) y ) ) C_ ran F
98	65, 97	unssi 3675	. . . . 5 |- ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F
99		fiss 7902	. . . . 5 |- ( ( ran F e. _V /\ ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F ) -> ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F ) )
100	24, 98, 99	mp2an 672	. . . 4 |- ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F )
101		tgss 19597	. . . 4 |- ( ( ( fi ` ran F ) e. _V /\ ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F ) ) -> ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) ) C_ ( topGen ` ( fi ` ran F ) ) )
102	4, 100, 101	mp2an 672	. . 3 |- ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) ) C_ ( topGen ` ( fi ` ran F ) )
103	3, 102	eqsstri 3529	. 2 |- (ordTop ` <_ ) C_ ( topGen ` ( fi ` ran F ) )
104		letop 19834	. . 3 |- (ordTop ` <_ ) e. Top
105		tgfiss 19620	. . 3 |- ( ( (ordTop ` <_ ) e. Top /\ ran F C_ (ordTop ` <_ ) ) -> ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ ) )
106	104, 23, 105	mp2an 672	. 2 |- ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ )
107	103, 106	eqssi 3515	1 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   E.wrex 2808   _Vcvv 3109   \ cdif 3468   u. cun 3469   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   class class class wbr 4456   |-> cmpt 4515   X. cxp 5006   ran crn 5009   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   ficfi 7888   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644   < clt 9645   <_ cle 9646   (,]cioc 11555   [,)cico 11556   [,]cicc 11557   topGenctg 14855  ordTopcordt 14916   Topctop 19521   Clsdccld 19644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ioc 11559  df-ico 11560  df-icc 11561  df-topgen 14861  df-ordt 14918  df-ps 15957  df-tsr 15958  df-top 19526  df-bases 19528  df-topon 19529  df-cld 19647

This theorem is referenced by: (None)