Theory: TOPOLOGY

Parents


Type constants


Term constants


Axioms


Definitions

re_Union
|- !S'. re_Union S' = (\x. ?s. S' s /\ s x)
re_union
|- !P Q. P re_union Q = (\x. P x \/ Q x)
re_intersect
|- !P Q. P re_intersect Q = (\x. P x /\ Q x)
re_null
|- re_null = (\x. F)
re_universe
|- re_universe = (\x. T)
re_subset
|- !P Q. P re_subset Q = (!x. P x ==> Q x)
re_compl
|- !S'. re_compl S' = (\x. ~(S' x))
istopology
|- !L.
     istopology L =
     L re_null /\
     L re_universe /\
     (!a b. L a /\ L b ==> L (a re_intersect b)) /\
     (!P. P re_subset L ==> L (re_Union P))
topology_TY_DEF
|- ?rep. TYPE_DEFINITION istopology rep
topology_tybij
|- (!a. topology (open a) = a) /\ (!r. istopology r = open (topology r) = r)
neigh
|- !top N x. neigh top (N,x) = (?P. open top P /\ P re_subset N /\ P x)
closed
|- !L S'. closed L S' = open L (re_compl S')
limpt
|- !top x S'.
     limpt top x S' = (!N. neigh top (N,x) ==> (?y. ~(x = y) /\ S' y /\ N y))
ismet
|- !m.
     ismet m =
     (!x y. (m (x,y) = & 0) = x = y) /\
     (!x y z. m (y,z) |<=| m (x,y) |+| m (x,z))
metric_TY_DEF
|- ?rep. TYPE_DEFINITION ismet rep
metric_tybij
|- (!a. metric (dist a) = a) /\ (!r. ismet r = dist (metric r) = r)
mtop
|- !m.
     mtop m =
     topology
       (\S'.
         !x. S' x ==> (?e. & 0 |<| e /\ (!y. dist m (x,y) |<| e ==> S' y)))
ball
|- !m x e. B m (x,e) = (\y. dist m (x,y) |<| e)
mr1
|- mr1 = metric (\(x,y). abs (y |-| x))

Theorems

SUBSET_REFL
|- !S'. S' re_subset S'
COMPL_MEM
|- !S' x. S' x = ~(re_compl S' x)
SUBSET_ANTISYM
|- !P Q. P re_subset Q /\ Q re_subset P = P = Q
SUBSET_TRANS
|- !P Q R. P re_subset Q /\ Q re_subset R ==> P re_subset R
TOPOLOGY
|- !L.
     open L re_null /\
     open L re_universe /\
     (!x y. open L x /\ open L y ==> open L (x re_intersect y)) /\
     (!P. P re_subset open L ==> open L (re_Union P))
TOPOLOGY_UNION
|- !L P. P re_subset open L ==> open L (re_Union P)
OPEN_OWN_NEIGH
|- !S' top x. open top S' /\ S' x ==> neigh top (S',x)
OPEN_UNOPEN
|- !S' top. open top S' = re_Union (\P. open top P /\ P re_subset S') = S'
OPEN_SUBOPEN
|- !S' top.
     open top S' = (!x. S' x ==> (?P. P x /\ open top P /\ P re_subset S'))
OPEN_NEIGH
|- !S' top.
     open top S' = (!x. S' x ==> (?N. neigh top (N,x) /\ N re_subset S'))
CLOSED_LIMPT
|- !top S'. closed top S' = (!x. limpt top x S' ==> S' x)
METRIC_ISMET
|- !m. ismet (dist m)
METRIC_ZERO
|- !m x y. (dist m (x,y) = & 0) = x = y
METRIC_SAME
|- !m x. dist m (x,x) = & 0
METRIC_POS
|- !m x y. & 0 |<=| dist m (x,y)
METRIC_SYM
|- !m x y. dist m (x,y) = dist m (y,x)
METRIC_TRIANGLE
|- !m x y z. dist m (x,z) |<=| dist m (x,y) |+| dist m (y,z)
METRIC_NZ
|- !m x y. ~(x = y) ==> & 0 |<| dist m (x,y)
mtop_istopology
|- !m.
     istopology
       (\S'.
         !x. S' x ==> (?e. & 0 |<| e /\ (!y. dist m (x,y) |<| e ==> S' y)))
MTOP_OPEN
|- !S' m.
     open (mtop m) S' =
     (!x. S' x ==> (?e. & 0 |<| e /\ (!y. dist m (x,y) |<| e ==> S' y)))
BALL_OPEN
|- !m x e. & 0 |<| e ==> open (mtop m) (B m (x,e))
BALL_NEIGH
|- !m x e. & 0 |<| e ==> neigh (mtop m) (B m (x,e),x)
MTOP_LIMPT
|- !m x S'.
     limpt (mtop m) x S' =
     (!e. & 0 |<| e ==> (?y. ~(x = y) /\ S' y /\ dist m (x,y) |<| e))
ISMET_R1
|- ismet (\(x,y). abs (y |-| x))
MR1_DEF
|- !x y. dist mr1 (x,y) = abs (y |-| x)
MR1_ADD
|- !x d. dist mr1 (x,x |+| d) = abs d
MR1_SUB
|- !x d. dist mr1 (x,x |-| d) = abs d
MR1_ADD_POS
|- !x d. & 0 |<=| d ==> (dist mr1 (x,x |+| d) = d)
MR1_SUB_LE
|- !x d. & 0 |<=| d ==> (dist mr1 (x,x |-| d) = d)
MR1_ADD_LT
|- !x d. & 0 |<| d ==> (dist mr1 (x,x |+| d) = d)
MR1_SUB_LT
|- !x d. & 0 |<| d ==> (dist mr1 (x,x |-| d) = d)
MR1_BETWEEN1
|- !x y z. x |<| z /\ dist mr1 (x,y) |<| z |-| x ==> y |<| z
MR1_LIMPT
|- !x. limpt (mtop mr1) x re_universe