Theorem lecldbas 17237

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 2404	. . . 4 |- ran ( y e. RR* |-> ( y (,] +oo ) ) = ran ( y e. RR* |-> ( y (,] +oo ) )
2		eqid 2404	. . . 4 |- ran ( y e. RR* |-> ( -oo [,) y ) ) = ran ( y e. RR* |-> ( -oo [,) y ) )
3	1, 2	leordtval2 17230	. . 3 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) )
4		fvex 5701	. . . 4 |- ( fi ` ran F ) e. _V
5		fvex 5701	. . . . . 6 |- (ordTop ` <_ ) e. _V
6		lecldbas.1	. . . . . . . 8 |- F = ( x e. ran [,] |-> ( RR* \ x ) )
7		iccf 10959	. . . . . . . . . . 11 |- [,] : ( RR* X. RR* ) --> ~P RR*
8		ffn 5550	. . . . . . . . . . 11 |- ( [,] : ( RR* X. RR* ) --> ~P RR* -> [,] Fn ( RR* X. RR* ) )
9	7, 8	ax-mp 8	. . . . . . . . . 10 |- [,] Fn ( RR* X. RR* )
10		ovelrn 6181	. . . . . . . . . 10 |- ( [,] Fn ( RR* X. RR* ) -> ( x e. ran [,] <-> E. a e. RR* E. b e. RR* x = ( a [,] b ) ) )
11	9, 10	ax-mp 8	. . . . . . . . 9 |- ( x e. ran [,] <-> E. a e. RR* E. b e. RR* x = ( a [,] b ) )
12		difeq2 3419	. . . . . . . . . . . 12 |- ( x = ( a [,] b ) -> ( RR* \ x ) = ( RR* \ ( a [,] b ) ) )
13		iccordt 17232	. . . . . . . . . . . . 13 |- ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) )
14		letopuni 17225	. . . . . . . . . . . . . 14 |- RR* = U. (ordTop ` <_ )
15	14	cldopn 17050	. . . . . . . . . . . . 13 |- ( ( a [,] b ) e. ( Clsd ` (ordTop ` <_ ) ) -> ( RR* \ ( a [,] b ) ) e. (ordTop ` <_ ) )
16	13, 15	ax-mp 8	. . . . . . . . . . . 12 |- ( RR* \ ( a [,] b ) ) e. (ordTop ` <_ )
17	12, 16	syl6eqel 2492	. . . . . . . . . . 11 |- ( x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
18	17	rexlimivw 2786	. . . . . . . . . 10 |- ( E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
19	18	rexlimivw 2786	. . . . . . . . 9 |- ( E. a e. RR* E. b e. RR* x = ( a [,] b ) -> ( RR* \ x ) e. (ordTop ` <_ ) )
20	11, 19	sylbi 188	. . . . . . . 8 |- ( x e. ran [,] -> ( RR* \ x ) e. (ordTop ` <_ ) )
21	6, 20	fmpti 5851	. . . . . . 7 |- F : ran [,] --> (ordTop ` <_ )
22		frn 5556	. . . . . . 7 |- ( F : ran [,] --> (ordTop ` <_ ) -> ran F C_ (ordTop ` <_ ) )
23	21, 22	ax-mp 8	. . . . . 6 |- ran F C_ (ordTop ` <_ )
24	5, 23	ssexi 4308	. . . . 5 |- ran F e. _V
25		eqid 2404	. . . . . . . 8 |- ( y e. RR* |-> ( y (,] +oo ) ) = ( y e. RR* |-> ( y (,] +oo ) )
26		mnfxr 10670	. . . . . . . . . . 11 |- -oo e. RR*
27		fnovrn 6180	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ -oo e. RR* /\ y e. RR* ) -> ( -oo [,] y ) e. ran [,] )
28	9, 26, 27	mp3an12 1269	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,] y ) e. ran [,] )
29	26	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo e. RR* )
30		id 20	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y e. RR* )
31		pnfxr 10669	. . . . . . . . . . . . . . 15 |- +oo e. RR*
32	31	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. RR* -> +oo e. RR* )
33		mnfle 10685	. . . . . . . . . . . . . 14 |- ( y e. RR* -> -oo <_ y )
34		pnfge 10683	. . . . . . . . . . . . . 14 |- ( y e. RR* -> y <_ +oo )
35		df-icc 10879	. . . . . . . . . . . . . . 15 |- [,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c <_ b ) } )
36		df-ioc 10877	. . . . . . . . . . . . . . 15 |- (,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a < c /\ c <_ b ) } )
37		xrltnle 9100	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y < z <-> -. z <_ y ) )
38		xrletr 10704	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z <_ y /\ y <_ +oo ) -> z <_ +oo ) )
39		xrlelttr 10702	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo < z ) )
40		xrltle 10698	. . . . . . . . . . . . . . . . 17 |- ( ( -oo e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
41	40	3adant2 976	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( -oo < z -> -oo <_ z ) )
42	39, 41	syld 42	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y < z ) -> -oo <_ z ) )
43	35, 36, 37, 35, 38, 42	ixxun 10888	. . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = ( -oo [,] +oo ) )
45		iccmax 10942	. . . . . . . . . . . . 13 |- ( -oo [,] +oo ) = RR*
46	44, 45	syl6eq 2452	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* )
47		iccssxr 10949	. . . . . . . . . . . . 13 |- ( -oo [,] y ) C_ RR*
48	35, 36, 37	ixxdisj 10887	. . . . . . . . . . . . . 14 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
49	26, 31, 48	mp3an13 1270	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,] y ) i^i ( y (,] +oo ) ) = (/) )
50		uneqdifeq 3676	. . . . . . . . . . . . 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 645	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( ( -oo [,] y ) u. ( y (,] +oo ) ) = RR* <-> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) ) )
52	46, 51	mpbid 202	. . . . . . . . . . 11 |- ( y e. RR* -> ( RR* \ ( -oo [,] y ) ) = ( y (,] +oo ) )
53	52	eqcomd 2409	. . . . . . . . . 10 |- ( y e. RR* -> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) )
54		difeq2 3419	. . . . . . . . . . . 12 |- ( x = ( -oo [,] y ) -> ( RR* \ x ) = ( RR* \ ( -oo [,] y ) ) )
55	54	eqeq2d 2415	. . . . . . . . . . 11 |- ( x = ( -oo [,] y ) -> ( ( y (,] +oo ) = ( RR* \ x ) <-> ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) )
56	55	rspcev 3012	. . . . . . . . . 10 |- ( ( ( -oo [,] y ) e. ran [,] /\ ( y (,] +oo ) = ( RR* \ ( -oo [,] y ) ) ) -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
57	28, 53, 56	syl2anc 643	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
58		xrex 10565	. . . . . . . . . . 11 |- RR* e. _V
59		difexg 4311	. . . . . . . . . . 11 |- ( RR* e. _V -> ( RR* \ x ) e. _V )
60	58, 59	ax-mp 8	. . . . . . . . . 10 |- ( RR* \ x ) e. _V
61	6, 60	elrnmpti 5080	. . . . . . . . 9 |- ( ( y (,] +oo ) e. ran F <-> E. x e. ran [,] ( y (,] +oo ) = ( RR* \ x ) )
62	57, 61	sylibr 204	. . . . . . . 8 |- ( y e. RR* -> ( y (,] +oo ) e. ran F )
63	25, 62	fmpti 5851	. . . . . . 7 |- ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F
64		frn 5556	. . . . . . 7 |- ( ( y e. RR* |-> ( y (,] +oo ) ) : RR* --> ran F -> ran ( y e. RR* |-> ( y (,] +oo ) ) C_ ran F )
65	63, 64	ax-mp 8	. . . . . 6 |- ran ( y e. RR* |-> ( y (,] +oo ) ) C_ ran F
66		eqid 2404	. . . . . . . 8 |- ( y e. RR* |-> ( -oo [,) y ) ) = ( y e. RR* |-> ( -oo [,) y ) )
67		fnovrn 6180	. . . . . . . . . . 11 |- ( ( [,] Fn ( RR* X. RR* ) /\ y e. RR* /\ +oo e. RR* ) -> ( y [,] +oo ) e. ran [,] )
68	9, 31, 67	mp3an13 1270	. . . . . . . . . 10 |- ( y e. RR* -> ( y [,] +oo ) e. ran [,] )
69		df-ico 10878	. . . . . . . . . . . . . . 15 |- [,) = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c < b ) } )
70		xrlenlt 9099	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ z e. RR* ) -> ( y <_ z <-> -. z < y ) )
71		xrltletr 10703	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z < +oo ) )
72		xrltle 10698	. . . . . . . . . . . . . . . . 17 |- ( ( z e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
73	72	3adant2 976	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( z < +oo -> z <_ +oo ) )
74	71, 73	syld 42	. . . . . . . . . . . . . . 15 |- ( ( z e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( z < y /\ y <_ +oo ) -> z <_ +oo ) )
75		xrletr 10704	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ z e. RR* ) -> ( ( -oo <_ y /\ y <_ z ) -> -oo <_ z ) )
76	69, 35, 70, 35, 74, 75	ixxun 10888	. . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( -oo [,] +oo ) )
78		uncom 3451	. . . . . . . . . . . . 13 |- ( ( -oo [,) y ) u. ( y [,] +oo ) ) = ( ( y [,] +oo ) u. ( -oo [,) y ) )
79	77, 78, 45	3eqtr3g 2459	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* )
80		iccssxr 10949	. . . . . . . . . . . . 13 |- ( y [,] +oo ) C_ RR*
81		incom 3493	. . . . . . . . . . . . . 14 |- ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = ( ( -oo [,) y ) i^i ( y [,] +oo ) )
82	69, 35, 70	ixxdisj 10887	. . . . . . . . . . . . . . 15 |- ( ( -oo e. RR* /\ y e. RR* /\ +oo e. RR* ) -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
83	26, 31, 82	mp3an13 1270	. . . . . . . . . . . . . 14 |- ( y e. RR* -> ( ( -oo [,) y ) i^i ( y [,] +oo ) ) = (/) )
84	81, 83	syl5eq 2448	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( ( y [,] +oo ) i^i ( -oo [,) y ) ) = (/) )
85		uneqdifeq 3676	. . . . . . . . . . . . 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 645	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( ( y [,] +oo ) u. ( -oo [,) y ) ) = RR* <-> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) ) )
87	79, 86	mpbid 202	. . . . . . . . . . 11 |- ( y e. RR* -> ( RR* \ ( y [,] +oo ) ) = ( -oo [,) y ) )
88	87	eqcomd 2409	. . . . . . . . . 10 |- ( y e. RR* -> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) )
89		difeq2 3419	. . . . . . . . . . . 12 |- ( x = ( y [,] +oo ) -> ( RR* \ x ) = ( RR* \ ( y [,] +oo ) ) )
90	89	eqeq2d 2415	. . . . . . . . . . 11 |- ( x = ( y [,] +oo ) -> ( ( -oo [,) y ) = ( RR* \ x ) <-> ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) )
91	90	rspcev 3012	. . . . . . . . . 10 |- ( ( ( y [,] +oo ) e. ran [,] /\ ( -oo [,) y ) = ( RR* \ ( y [,] +oo ) ) ) -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
92	68, 88, 91	syl2anc 643	. . . . . . . . 9 |- ( y e. RR* -> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
93	6, 60	elrnmpti 5080	. . . . . . . . 9 |- ( ( -oo [,) y ) e. ran F <-> E. x e. ran [,] ( -oo [,) y ) = ( RR* \ x ) )
94	92, 93	sylibr 204	. . . . . . . 8 |- ( y e. RR* -> ( -oo [,) y ) e. ran F )
95	66, 94	fmpti 5851	. . . . . . 7 |- ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F
96		frn 5556	. . . . . . 7 |- ( ( y e. RR* |-> ( -oo [,) y ) ) : RR* --> ran F -> ran ( y e. RR* |-> ( -oo [,) y ) ) C_ ran F )
97	95, 96	ax-mp 8	. . . . . 6 |- ran ( y e. RR* |-> ( -oo [,) y ) ) C_ ran F
98	65, 97	unssi 3482	. . . . 5 |- ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) C_ ran F
99		fiss 7387	. . . . 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 654	. . . 4 |- ( fi ` ( ran ( y e. RR* |-> ( y (,] +oo ) ) u. ran ( y e. RR* |-> ( -oo [,) y ) ) ) ) C_ ( fi ` ran F )
101		tgss 16988	. . . 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 654	. . 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 3338	. 2 |- (ordTop ` <_ ) C_ ( topGen ` ( fi ` ran F ) )
104		letop 17224	. . 3 |- (ordTop ` <_ ) e. Top
105		tgfiss 17011	. . 3 |- ( ( (ordTop ` <_ ) e. Top /\ ran F C_ (ordTop ` <_ ) ) -> ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ ) )
106	104, 23, 105	mp2an 654	. 2 |- ( topGen ` ( fi ` ran F ) ) C_ (ordTop ` <_ )
107	103, 106	eqssi 3324	1 |- (ordTop ` <_ ) = ( topGen ` ( fi ` ran F ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   E.wrex 2667   _Vcvv 2916   \ cdif 3277   u. cun 3278   i^i cin 3279   C_ wss 3280   (/)c0 3588   ~Pcpw 3759   class class class wbr 4172   e. cmpt 4226   X. cxp 4835   ran crn 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   ficfi 7373   +oocpnf 9073   -oocmnf 9074   RR*cxr 9075   < clt 9076   <_ cle 9077   (,]cioc 10873   [,)cico 10874   [,]cicc 10875   topGenctg 13620  ordTopcordt 13676   Topctop 16913   Clsdccld 17035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-ioc 10877  df-ico 10878  df-icc 10879  df-topgen 13622  df-ordt 13680  df-ps 14584  df-tsr 14585  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038