Theorem lecldbas 18723

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 2441	. . . 4 |- ran ( y e. RR* |-> ( y (,] +oo ) ) = ran ( y e. RR* |-> ( y (,] +oo ) )
2		eqid 2441	. . . 4 |- ran ( y e. RR* |-> ( -oo [,) y ) ) = ran ( y e. RR* |-> ( -oo [,) y ) )
3	1, 2	leordtval2 18716	. . 3 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) )
4		fvex 5698	. . . 4 |- ( fi ` ran F ) e. _V
5		fvex 5698	. . . . . 6 |- (ordTop ` <_ ) e. _V
6		lecldbas.1	. . . . . . . 8 |- F = ( x e. ran [,] |-> ( RR* \ x ) )
7		iccf 11384	. . . . . . . . . . 11 |- [,] : ( RR* X. RR* ) --> ~P RR*
8		ffn 5556	. . . . . . . . . . 11 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> [,] Fn ( RR* X. RR* ) )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- [,] Fn ( RR* X. RR* )
10		ovelrn 6238	. . . . . . . . . 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 3465	. . . . . . . . . . . 12 |- ( x = ( a [,] b ) -> ( RR* \ x ) = ( RR* \ ( a [,] b ) ) )
13		iccordt 18718	. . . . . . . . . . . . 13 |- ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) )
14		letopuni 18711	. . . . . . . . . . . . . 14 |- RR* = U. (ordTop ` <_ )
15	14	cldopn 18535	. . . . . . . . . . . . 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 2529	. . . . . . . . . . 11 |- ( x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
18	17	rexlimivw 2835	. . . . . . . . . 10 |- ( E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
19	18	rexlimivw 2835	. . . . . . . . 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 5863	. . . . . . 7 |- F : ran [,] --> (ordTop ` <_ )
22		frn 5562	. . . . . . 7 |- ( F : ran [,] --> (ordTop ` <_ ) -> ran F C_ (ordTop ` <_ ) )
23	21, 22	ax-mp 5	. . . . . 6 |- ran F C_ (ordTop ` <_ )
24	5, 23	ssexi 4434	. . . . 5 |- ran F e. _V
25		eqid 2441	. . . . . . . 8 |- ( y e. RR* |-> ( y (,] +oo ) ) = ( y e. RR* |-> ( y (,] +oo ) )
26		mnfxr 11090	. . . . . . . . . . 11 |- -oo e. RR*
27		fnovrn 6237	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ -oo e. RR* /\ y e. RR* ) -> ( -oo [,] y ) e. ran [,] )
28	9, 26, 27	mp3an12 1299	. . . . . . . . . 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 11088	. . . . . . . . . . . . . . 15 |- +oo e. RR*
32	31	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> +oo e. RR* )
33		mnfle 11109	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo <_ y )
34		pnfge 11106	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y <_ +oo )
35		df-icc 11303	. . . . . . . . . . . . . . 15 |- [,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c <_ b ) } )
36		df-ioc 11301	. . . . . . . . . . . . . . 15 |- (,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a < c /\ c <_ b ) } )
37		xrltnle 9439	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y < z <-> -. z <_ y ) )
38		xrletr 11128	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z <_ y /\ y <_ +oo ) -> z <_ +oo ) )
39		xrlelttr 11126	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo < z ) )
40		xrltle 11122	. . . . . . . . . . . . . . . . 17 |- ( ( -oo e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
41	40	3adant2 1002	. . . . . . . . . . . . . . . 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 11312	. . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = ( -oo [,] +oo ) )
45		iccmax 11367	. . . . . . . . . . . . 13 |- ( -oo [,] +oo ) = RR*
46	44, 45	syl6eq 2489	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* )
47		iccssxr 11374	. . . . . . . . . . . . 13 |- ( -oo [,] y ) C_ RR*
48	35, 36, 37	ixxdisj 11311	. . . . . . . . . . . . . 14 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
49	26, 31, 48	mp3an13 1300	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
50		uneqdifeq 3764	. . . . . . . . . . . . 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 658	. . . . . . . . . . . 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 2446	. . . . . . . . . 10 |- ( y e. RR* -> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) )
54		difeq2 3465	. . . . . . . . . . . 12 |- ( x = ( -oo [,] y ) -> ( RR* \ x ) = ( RR* \ ( -oo [,] y ) ) )
55	54	eqeq2d 2452	. . . . . . . . . . 11 |- ( x = ( -oo [,] y ) -> ( ( y (,] +oo ) = ( RR* \ x ) <-> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) )
56	55	rspcev 3070	. . . . . . . . . 10 |- ( ( ( -oo [,] y ) e. ran [,] /\ ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
57	28, 53, 56	syl2anc 656	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
58		xrex 10984	. . . . . . . . . . 11 |- RR* e. _V
59		difexg 4437	. . . . . . . . . . 11 |- ( RR* e. _V -> ( RR* \ x ) e. _V )
60	58, 59	ax-mp 5	. . . . . . . . . 10 |- ( RR* \ x ) e. _V
61	6, 60	elrnmpti 5086	. . . . . . . . 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 5863	. . . . . . 7 |- ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F
64		frn 5562	. . . . . . 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 2441	. . . . . . . 8 |- ( y e. RR* |-> ( -oo [,) y ) ) = ( y e. RR* |-> ( -oo [,) y ) )
67		fnovrn 6237	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ y e. RR* /\ +oo e. RR* ) -> ( y [,] +oo ) e. ran [,] )
68	9, 31, 67	mp3an13 1300	. . . . . . . . . 10 |- ( y e. RR* -> ( y [,] +oo ) e. ran [,] )
69		df-ico 11302	. . . . . . . . . . . . . . 15 |- [,) = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c < b ) } )
70		xrlenlt 9438	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y <_ z <-> -. z < y ) )
71		xrltletr 11127	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z < +oo ) )
72		xrltle 11122	. . . . . . . . . . . . . . . . 17 |- ( ( z e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
73	72	3adant2 1002	. . . . . . . . . . . . . . . 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 11128	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y <_ z ) -> -oo <_ z ) )
76	69, 35, 70, 35, 74, 75	ixxun 11312	. . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( -oo [,] +oo ) )
78		uncom 3497	. . . . . . . . . . . . 13 |- ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( ( y [,] +oo ) u. ( -oo [,) y ) )
79	77, 78, 45	3eqtr3g 2496	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* )
80		iccssxr 11374	. . . . . . . . . . . . 13 |- ( y [,] +oo ) C_ RR*
81		incom 3540	. . . . . . . . . . . . . 14 |- ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = ( ( -oo [,) y ) i^i ( y [,] +oo ) )
82	69, 35, 70	ixxdisj 11311	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
83	26, 31, 82	mp3an13 1300	. . . . . . . . . . . . . 14 |- ( y e. RR* -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
84	81, 83	syl5eq 2485	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) )
85		uneqdifeq 3764	. . . . . . . . . . . . 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 658	. . . . . . . . . . . 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 2446	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) )
89		difeq2 3465	. . . . . . . . . . . 12 |- ( x = ( y [,] +oo ) -> ( RR* \ x ) = ( RR* \ ( y [,] +oo ) ) )
90	89	eqeq2d 2452	. . . . . . . . . . 11 |- ( x = ( y [,] +oo ) -> ( ( -oo [,) y ) = ( RR* \ x ) <-> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) )
91	90	rspcev 3070	. . . . . . . . . 10 |- ( ( ( y [,] +oo ) e. ran [,] /\ ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
92	68, 88, 91	syl2anc 656	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
93	6, 60	elrnmpti 5086	. . . . . . . . 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 5863	. . . . . . 7 |- ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F
96		frn 5562	. . . . . . 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 3528	. . . . 5 |- ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F
99		fiss 7670	. . . . 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 667	. . . 4 |- ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F )
101		tgss 18473	. . . 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 667	. . 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 3383	. 2 |- (ordTop ` <_ ) C_ ( topGen ` ( fi ` ran F ) )
104		letop 18710	. . 3 |- (ordTop ` <_ ) e. Top
105		tgfiss 18496	. . 3 |- ( ( (ordTop ` <_ ) e. Top /\ ran F C_ (ordTop ` <_ ) ) -> ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ ) )
106	104, 23, 105	mp2an 667	. 2 |- ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ )
107	103, 106	eqssi 3369	1 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   E.wrex 2714   _Vcvv 2970   \ cdif 3322   u. cun 3323   i^i cin 3324   C_ wss 3325   (/)c0 3634   ~Pcpw 3857   class class class wbr 4289   e. cmpt 4347   X. cxp 4834   ran crn 4837   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   ficfi 7656   +oocpnf 9411   -oocmnf 9412   RR*cxr 9413   < clt 9414   <_ cle 9415   (,]cioc 11297   [,)cico 11298   [,]cicc 11299   topGenctg 14372  ordTopcordt 14433   Topctop 18398   Clsdccld 18520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-ioc 11301  df-ico 11302  df-icc 11303  df-topgen 14378  df-ordt 14435  df-ps 15366  df-tsr 15367  df-top 18403  df-bases 18405  df-topon 18406  df-cld 18523

This theorem is referenced by: (None)