:: Complex Spaces
:: by Czes{\l}aw Byli\'nski and Andrzej Trybulec
::
:: Received September 27, 1990
:: Copyright (c) 1990-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, COMPLEX1, ARYTM_3, ARYTM_1, TARSKI, XBOOLE_0,
CARD_1, RCOMP_1, SETFAM_1, METRIC_1, PRE_TOPC, STRUCT_0, CARD_5,
COMPLSP1;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0,
XREAL_0, SETFAM_1, REAL_1, COMPLEX1, FUNCT_1, RELSET_1, FUNCT_2,
STRUCT_0, PRE_TOPC, FINSEQ_1, VALUED_1, RVSUM_1, FINSEQ_2, SEQ_4;
constructors SETFAM_1, PARTFUN1, BINOP_1, SETWISEO, REAL_1, SQUARE_1, BINOP_2,
COMPLEX1, SEQ_4, FINSEQOP, RVSUM_1, COMPTS_1, XXREAL_2, RELSET_1;
registrations NUMBERS, XREAL_0, MEMBERED, FINSEQ_2, PRE_TOPC;
requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM;
begin
reserve n for Element of NAT,
x for Element of COMPLEX n;
definition let n;
func the_Complex_Space n -> strict TopSpace equals
:: COMPLSP1:def 1
TopStruct(#COMPLEX n,ComplexOpenSets(n)#);
end;
registration
let n;
cluster the_Complex_Space n -> non empty;
end;
theorem :: COMPLSP1:1
the topology of the_Complex_Space n = ComplexOpenSets n;
theorem :: COMPLSP1:2
the carrier of the_Complex_Space n = COMPLEX n;
reserve p,q for Point of the_Complex_Space n,
V for Subset of the_Complex_Space n;
theorem :: COMPLSP1:3
p is Element of COMPLEX n;
theorem :: COMPLSP1:4
for A being Subset of COMPLEX n st A = V holds A is open iff V is open;
theorem :: COMPLSP1:5
for A being Subset of COMPLEX n st A = V holds A is closed iff V is closed;
theorem :: COMPLSP1:6
the_Complex_Space n is T_2;
theorem :: COMPLSP1:7
the_Complex_Space n is regular;