Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Fix Point Theorem for Compact Spaces

**by****Alicia de la Cruz**- Received July 17, 1991
- MML identifier: ALI2

- [ Mizar article, MML identifier index ]

environ vocabulary METRIC_1, FINSET_1, BOOLE, ZFMISC_1, FUNCT_1, PRE_TOPC, FUNCOP_1, RELAT_1, ARYTM_3, PCOMPS_1, COMPTS_1, SETFAM_1, POWER, SUBSET_1, ARYTM_1, SEQ_1, SEQ_2, ORDINAL2, SEQM_3, ALI2, HAHNBAN; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XREAL_0, FINSET_1, SETFAM_1, METRIC_1, RELAT_1, FUNCT_2, STRUCT_0, PRE_TOPC, POWER, FUNCOP_1, COMPTS_1, PCOMPS_1, TOPS_2, SEQ_1, SEQ_2, SEQM_3, REAL_1, NAT_1; constructors FINSET_1, POWER, FUNCOP_1, COMPTS_1, PCOMPS_1, TOPS_2, SEQ_2, SEQM_3, REAL_1, NAT_1, PARTFUN1, MEMBERED; clusters STRUCT_0, PCOMPS_1, FUNCOP_1, XREAL_0, SEQ_1, METRIC_1, MEMBERED, ZFMISC_1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve M for non empty MetrSpace; theorem :: ALI2:1 for F being set st F is finite & F <> {} & F is c=-linear ex m being set st m in F & for C being set st C in F holds m c= C; definition let M be non empty MetrSpace; mode contraction of M -> Function of the carrier of M, the carrier of M means :: ALI2:def 1 ex L being Real st 0 < L & L < 1 & for x,y being Point of M holds dist(it.x,it.y) <= L*dist(x,y); end; theorem :: ALI2:2 for f being contraction of M st TopSpaceMetr(M) is compact ex c being Point of M st f.c =c & :: exists a fix point for x being Point of M st f.x=x holds x=c;

Back to top