Nuprl Lemma : punctured-homeomorphism
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)].
(mcomplete(X with d)
⇒ mcomplete(Y with d')
⇒ (∀h:homeomorphic(X;d;Y;d'). ∀p:Y. (h ∈ homeomorphic({x:X| x # (snd(h)) p} ;d;{y:Y| y # p} ;d'))))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
homeomorphic: homeomorphic(X;dX;Y;dY),
mk-metric-space: X with d,
msep: x # y,
metric: metric(X),
uall: ∀[x:A]. B[x],
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
homeomorphic: homeomorphic(X;dX;Y;dY),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
pi2: snd(t),
mfun: FUN(X ⟶ Y),
and: P ∧ Q,
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
msfun: msfun(X;d;Y;d'),
squash: ↓T,
sq_stable: SqStable(P),
is-msfun: is-msfun(X;d;Y;d';f),
prop: ℙ,
uimplies: b supposing a,
cand: A c∧ B,
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
is-mfun: f:FUN(X;Y)
Latex:
\mforall{}[X,Y:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':metric(Y)].
(mcomplete(X with d)
{}\mRightarrow{} mcomplete(Y with d')
{}\mRightarrow{} (\mforall{}h:homeomorphic(X;d;Y;d'). \mforall{}p:Y. (h \mmember{} homeomorphic(\{x:X| x \# (snd(h)) p\} ;d;\{y:Y| y \# p\} ;d')\000C)))
Date html generated:
2020_05_20-AM-11_59_01
Last ObjectModification:
2019_11_11-PM-04_42_48
Theory : reals
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