Theorem lecldbas 19502

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 2467	. . . 4 |- ran ( y e. RR* |-> ( y (,] +oo ) ) = ran ( y e. RR* |-> ( y (,] +oo ) )
2		eqid 2467	. . . 4 |- ran ( y e. RR* |-> ( -oo [,) y ) ) = ran ( y e. RR* |-> ( -oo [,) y ) )
3	1, 2	leordtval2 19495	. . 3 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) )
4		fvex 5875	. . . 4 |- ( fi ` ran F ) e. _V
5		fvex 5875	. . . . . 6 |- (ordTop ` <_ ) e. _V
6		lecldbas.1	. . . . . . . 8 |- F = ( x e. ran [,] |-> ( RR* \ x ) )
7		iccf 11622	. . . . . . . . . . 11 |- [,] : ( RR* X. RR* ) --> ~P RR*
8		ffn 5730	. . . . . . . . . . 11 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> [,] Fn ( RR* X. RR* ) )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- [,] Fn ( RR* X. RR* )
10		ovelrn 6434	. . . . . . . . . 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 3616	. . . . . . . . . . . 12 |- ( x = ( a [,] b ) -> ( RR* \ x ) = ( RR* \ ( a [,] b ) ) )
13		iccordt 19497	. . . . . . . . . . . . 13 |- ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) )
14		letopuni 19490	. . . . . . . . . . . . . 14 |- RR* = U. (ordTop ` <_ )
15	14	cldopn 19314	. . . . . . . . . . . . 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 2563	. . . . . . . . . . 11 |- ( x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
18	17	rexlimivw 2952	. . . . . . . . . 10 |- ( E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
19	18	rexlimivw 2952	. . . . . . . . 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 6043	. . . . . . 7 |- F : ran [,] --> (ordTop ` <_ )
22		frn 5736	. . . . . . 7 |- ( F : ran [,] --> (ordTop ` <_ ) -> ran F C_ (ordTop ` <_ ) )
23	21, 22	ax-mp 5	. . . . . 6 |- ran F C_ (ordTop ` <_ )
24	5, 23	ssexi 4592	. . . . 5 |- ran F e. _V
25		eqid 2467	. . . . . . . 8 |- ( y e. RR* |-> ( y (,] +oo ) ) = ( y e. RR* |-> ( y (,] +oo ) )
26		mnfxr 11322	. . . . . . . . . . 11 |- -oo e. RR*
27		fnovrn 6433	. . . . . . . . . . 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 11320	. . . . . . . . . . . . . . 15 |- +oo e. RR*
32	31	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> +oo e. RR* )
33		mnfle 11341	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo <_ y )
34		pnfge 11338	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y <_ +oo )
35		df-icc 11535	. . . . . . . . . . . . . . 15 |- [,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c <_ b ) } )
36		df-ioc 11533	. . . . . . . . . . . . . . 15 |- (,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a < c /\ c <_ b ) } )
37		xrltnle 9652	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y < z <-> -. z <_ y ) )
38		xrletr 11360	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z <_ y /\ y <_ +oo ) -> z <_ +oo ) )
39		xrlelttr 11358	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo < z ) )
40		xrltle 11354	. . . . . . . . . . . . . . . . 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 11544	. . . . . . . . . . . . . 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 11599	. . . . . . . . . . . . 13 |- ( -oo [,] +oo ) = RR*
46	44, 45	syl6eq 2524	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* )
47		iccssxr 11606	. . . . . . . . . . . . 13 |- ( -oo [,] y ) C_ RR*
48	35, 36, 37	ixxdisj 11543	. . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . 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 2475	. . . . . . . . . 10 |- ( y e. RR* -> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) )
54		difeq2 3616	. . . . . . . . . . . 12 |- ( x = ( -oo [,] y ) -> ( RR* \ x ) = ( RR* \ ( -oo [,] y ) ) )
55	54	eqeq2d 2481	. . . . . . . . . . 11 |- ( x = ( -oo [,] y ) -> ( ( y (,] +oo ) = ( RR* \ x ) <-> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) )
56	55	rspcev 3214	. . . . . . . . . 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 11216	. . . . . . . . . . 11 |- RR* e. _V
59		difexg 4595	. . . . . . . . . . 11 |- ( RR* e. _V -> ( RR* \ x ) e. _V )
60	58, 59	ax-mp 5	. . . . . . . . . 10 |- ( RR* \ x ) e. _V
61	6, 60	elrnmpti 5252	. . . . . . . . 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 6043	. . . . . . 7 |- ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F
64		frn 5736	. . . . . . 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 2467	. . . . . . . 8 |- ( y e. RR* |-> ( -oo [,) y ) ) = ( y e. RR* |-> ( -oo [,) y ) )
67		fnovrn 6433	. . . . . . . . . . 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 11534	. . . . . . . . . . . . . . 15 |- [,) = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c < b ) } )
70		xrlenlt 9651	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y <_ z <-> -. z < y ) )
71		xrltletr 11359	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z < +oo ) )
72		xrltle 11354	. . . . . . . . . . . . . . . . 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 11360	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y <_ z ) -> -oo <_ z ) )
76	69, 35, 70, 35, 74, 75	ixxun 11544	. . . . . . . . . . . . . 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 3648	. . . . . . . . . . . . 13 |- ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( ( y [,] +oo ) u. ( -oo [,) y ) )
79	77, 78, 45	3eqtr3g 2531	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* )
80		iccssxr 11606	. . . . . . . . . . . . 13 |- ( y [,] +oo ) C_ RR*
81		incom 3691	. . . . . . . . . . . . . 14 |- ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = ( ( -oo [,) y ) i^i ( y [,] +oo ) )
82	69, 35, 70	ixxdisj 11543	. . . . . . . . . . . . . . 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 2520	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) )
85		uneqdifeq 3915	. . . . . . . . . . . . 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 2475	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) )
89		difeq2 3616	. . . . . . . . . . . 12 |- ( x = ( y [,] +oo ) -> ( RR* \ x ) = ( RR* \ ( y [,] +oo ) ) )
90	89	eqeq2d 2481	. . . . . . . . . . 11 |- ( x = ( y [,] +oo ) -> ( ( -oo [,) y ) = ( RR* \ x ) <-> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) )
91	90	rspcev 3214	. . . . . . . . . 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 5252	. . . . . . . . 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 6043	. . . . . . 7 |- ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F
96		frn 5736	. . . . . . 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 3679	. . . . 5 |- ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F
99		fiss 7883	. . . . 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 19252	. . . 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 3534	. 2 |- (ordTop ` <_ ) C_ ( topGen ` ( fi ` ran F ) )
104		letop 19489	. . 3 |- (ordTop ` <_ ) e. Top
105		tgfiss 19275	. . 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 3520	1 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   E.wrex 2815   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   ran crn 5000   Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   ficfi 7869   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626   < clt 9627   <_ cle 9628   (,]cioc 11529   [,)cico 11530   [,]cicc 11531   topGenctg 14692  ordTopcordt 14753   Topctop 19177   Clsdccld 19299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7870  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-ioc 11533  df-ico 11534  df-icc 11535  df-topgen 14698  df-ordt 14755  df-ps 15686  df-tsr 15687  df-top 19182  df-bases 19184  df-topon 19185  df-cld 19302

This theorem is referenced by: (None)