Theory "topology"

Parents     real

Signature

Type Arity
metric 1
topology 1
Constant Type
B :'a metric -> 'a # real -> 'a -> bool
istopology :(('a -> bool) -> bool) -> bool
re_subset :('a -> bool) -> ('a -> bool) -> bool
mr1 :real metric
re_null :'a -> bool
ismet :('a # 'a -> real) -> bool
closed :'a topology -> ('a -> bool) -> bool
re_intersect :('a -> bool) -> ('a -> bool) -> 'a -> bool
dist :'a metric -> 'a # 'a -> real
limpt :'a topology -> 'a -> ('a -> bool) -> bool
re_compl :('a -> bool) -> 'a -> bool
topology :(('a -> bool) -> bool) -> 'a topology
neigh :'a topology -> ('a -> bool) # 'a -> bool
re_Union :(('a -> bool) -> bool) -> 'a -> bool
re_union :('a -> bool) -> ('a -> bool) -> 'a -> bool
metric :('a # 'a -> real) -> 'a metric
mtop :'a metric -> 'a topology
open :'a topology -> ('a -> bool) -> bool
re_universe :'a -> bool

Definitions

re_Union
|- !P. re_Union P = (\x. ?s. P 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
|- !P. re_compl P = (\x. ~P 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. topology$B m (x,e) = (\y. dist m (x,y) < e)
mr1
|- mr1 = metric (\(x,y). abs (y - x))


Theorems

SUBSET_REFL
|- !P. P re_subset P
COMPL_MEM
|- !P x. P x = ~re_compl P 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) (topology$B m (x,e))
BALL_NEIGH
|- !m x e. 0 < e ==> neigh (mtop m) (topology$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