:: Metric Spaces as Topological Spaces - Fundamental Concepts
:: by Agata Darmochwa{\l} and Yatsuka Nakamura
::
:: Received November 21, 1991
:: Copyright (c) 1991 Association of Mizar Users
theorem Th1: :: TOPMETR:1
theorem :: TOPMETR:2
canceled;
theorem :: TOPMETR:3
theorem Th4: :: TOPMETR:4
theorem Th5: :: TOPMETR:5
theorem :: TOPMETR:6
theorem :: TOPMETR:7
canceled;
theorem :: TOPMETR:8
canceled;
theorem :: TOPMETR:9
theorem :: TOPMETR:10
Lm1:
for M being MetrSpace holds MetrStruct(# the carrier of M,the distance of M #) is MetrSpace
:: deftheorem Def1 defines SubSpace TOPMETR:def 1 :
theorem :: TOPMETR:11
canceled;
theorem Th12: :: TOPMETR:12
theorem Th13: :: TOPMETR:13
:: deftheorem Def2 defines | TOPMETR:def 2 :
:: deftheorem Def3 defines Closed-Interval-MSpace TOPMETR:def 3 :
theorem Th14: :: TOPMETR:14
:: deftheorem Def4 defines being_ball-family TOPMETR:def 4 :
theorem Th15: :: TOPMETR:15
theorem Th16: :: TOPMETR:16
theorem :: TOPMETR:17
canceled;
theorem :: TOPMETR:18
canceled;
theorem Th19: :: TOPMETR:19
theorem :: TOPMETR:20
canceled;
theorem Th21: :: TOPMETR:21
theorem Th22: :: TOPMETR:22
:: deftheorem TOPMETR:def 5 :
canceled;
:: deftheorem Def6 defines compact TOPMETR:def 6 :
theorem :: TOPMETR:23
:: deftheorem defines R^1 TOPMETR:def 7 :
theorem Th24: :: TOPMETR:24
:: deftheorem defines Closed-Interval-TSpace TOPMETR:def 8 :
theorem Th25: :: TOPMETR:25
theorem Th26: :: TOPMETR:26
theorem Th27: :: TOPMETR:27
Lm2:
for a, b, r being real number st r >= 0 & a + r <= b holds
a <= b
theorem :: TOPMETR:28
theorem :: TOPMETR:29
theorem Th30: :: TOPMETR:30
theorem :: TOPMETR:31