Nuprl Lemma : homeomorphic-retraction
∀[X,A:Type]. ∀[d:metric(X)].
Retract(X ⟶ A) ⇒ (∀Y:Type. ∀d':metric(Y). ∀h:homeomorphic(X;d;Y;d'). Retract(Y ⟶ homeo-image(A;Y;d';h)))
supposing metric-subspace(X;d;A)
Proof
Definitions occuring in Statement :
homeo-image: homeo-image(A;Y;dY;h),
m-retraction: Retract(X ⟶ A),
metric-subspace: metric-subspace(X;d;A),
homeomorphic: homeomorphic(X;dX;Y;dY),
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
metric-subspace: metric-subspace(X;d;A),
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
m-retraction: Retract(X ⟶ A),
homeomorphic: homeomorphic(X;dX;Y;dY),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
mfun: FUN(X ⟶ Y),
homeo-image: homeo-image(A;Y;dY;h),
subtype_rel: A ⊆r B,
pi1: fst(t),
prop: ℙ,
cand: A c∧ B,
strong-subtype: strong-subtype(A;B),
is-mfun: f:FUN(X;Y),
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[X,A:Type]. \mforall{}[d:metric(X)].
Retract(X {}\mrightarrow{} A)
{}\mRightarrow{} (\mforall{}Y:Type. \mforall{}d':metric(Y). \mforall{}h:homeomorphic(X;d;Y;d'). Retract(Y {}\mrightarrow{} homeo-image(A;Y;d';h)))
supposing metric-subspace(X;d;A)
Date html generated:
2020_05_20-AM-11_52_59
Last ObjectModification:
2019_11_11-PM-02_23_45
Theory : reals
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