A Note on Semialgebraically Proper Maps
We prove that a semialgebraic map is semialgebraically proper if and only if it is proper. As an application of this assertion, we compare the semialgebraically proper actions with proper actions in a sense of Palais.
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2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508169192603648 |
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| author | Park, Dae Heui Парк, Де Ху |
| author_facet | Park, Dae Heui Парк, Де Ху |
| author_sort | Park, Dae Heui |
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| datestamp_date | 2019-12-05T10:26:31Z |
| description | We prove that a semialgebraic map is semialgebraically proper if and only if it is proper. As an application of this assertion, we compare the semialgebraically proper actions with proper actions in a sense of Palais. |
| first_indexed | 2026-03-24T02:20:56Z |
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| fulltext |
UDC 512.5
Dae Heui Park (College Natural Sci., Chonnam Nat. Univ., Gwangju, Korea)
NOTE ON SEMIALGEBRAICALLY PROPER MAPS*
ПРО НАПIВАЛГЕБРАЇЧНО ВЛАСНI ВIДОБРАЖЕННЯ
In this paper we prove that a semialgebraic map is semialgebraically proper if and only if it is proper. As an application
of this, we compare the semialgebraically proper actions with the proper actions in the sense of Palais.
Доведено, що напiвалгебраїчне вiдображення є напiвалгебраїчно власним тодi i тiльки тодi, коли воно є власним.
Як застосування цього факту, порiвнюються напiвалгебраїчно власнi дiї з власними дiями в сенсi Пале.
1. Introduction. A semialgebraic space is an object obtained by pasting finitely many semialgebraic
sets together along open (or closed) semialgebraic subsets. For this reason, the topologies of semial-
gebraic spaces are of no interest, so we will only treat the semialgebraic sets over the real numbers
in this paper.
A semialgebraic set is a subset of some Rn defined by finite number of polynomial equations
and inequalities. Throughout this paper we consider the semialgebraic sets in Rn equipped with the
subspace topology induced by the usual topology of Rn. A continuous map f : X → Y between
semialgebraic sets X ⊂ Rm and Y ⊂ Rn is called semialgebraic if its graph is a semialgebraic
set in Rm+n. Usually, semialgebraic map just a map, not necessarily continuous, whose graph is
semialgebraic. However, since all semialgebraic maps occurring in this paper are continuous, for
simplicity, we will assume that all semialgebraic maps are continuous.
The purpose of this paper is to find the equivalence conditions for semialgebraically proper maps.
Recall that a map is called proper if the preimage of any compact set is compact. Similarly, a
semialgebraic map f : X → Y is called semialgebraically proper if the preimage f−1(C) is compact
for every compact and semialgebraic subset C of Y. Since C should be semialgebraic in the definition,
this notion is weaker than the condition that f is proper. However, in Section 2, we prove that a
semialgebraic map is semialgebraically proper if and only if it is proper. Note that this notion of
semialgebraically proper is slightly different from that of Delfs and Knebusch in [3].
As an application of the above result, we also discuss semialgebraically proper actions of semi-
algebraic groups on semialgebraic sets. Let M be a semialgebraic set and let G be a semialgebraic
group. We say M is a semialgebraic G-set if the action θ : G ×M → M is semialgebraic. A
semialgebraic action of G on M is called semialgebraically proper if the map
ϑ∗ : G×M →M ×M, (g, x) 7→ (θ(g, x), x)
is semialgebraically proper. In Section 3 we compare the semialgebraically proper actions with the
proper actions in the sense of Palais [6].
2. Semialgebraically proper maps. We first gather some properties concerning semialgebraic
sets and maps without proofs. For the details, we refer the reader to [1] and [4].
The class of semialgebraic sets in Rn is the smallest collection of subsets containing all subsets
of the form {x ∈ Rn | p(x) > 0} for a real valued polynomial p(x) = p(x1, . . . , xn), which is stable
under finite union, finite intersection and complement.
* Was financially supported by Chonnam National University, 2008.
c© DAE HEUI PARK, 2014
1262 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
NOTE ON SEMIALGEBRAICALLY PROPER MAPS 1263
Proposition 2.1. (1) Every semialgebraic set has a finite number of path connected components,
which are also semialgebraic.
(2) Let X be a semialgebraic set. If A is a semialgebraic subset of X, then the closure A, the
complement Ac and the interior A◦ in X are semialgebraic.
(3) Composition of two semialgebraic maps is semialgebraic.
(4) Let f : X → Y be a semialgebraic map between semialgebraic sets. If A is a semialgebraic
subset of X, then its image f(A) is semialgebraic. If B is a semialgebraic subset of Y, then its
preimage f−1(B) is semialgebraic.
(5) Let f : X → Q and g : X → Y be semialgebraic. Assume f is surjective. If h : Q→ Y is a
continuous map such that h ◦ f = g, then h is semialgebraic.
(6) Let X be a semialgebraic set. If V is a neighborhood of a point x in X, then there is a
semialgebraic neighborhood U of X with x ∈ U ⊂ V.
As a sequence plays an important role in the category of metric spaces, a curve germ plays in
the semialgebraic category, see [1] or [2]. A curve germ in a semialgebraic set X is represented by
a semialgebraic map α : (0, ε] → X for some ε > 0. Two curve germs are considered the same if,
after possible reparameterization of the intervals, they agree on a common subinterval (0, δ] for some
δ > 0. Thus, a curve germ is determined by the collection of images sets α((0, ε]) ⊂ X for ε > 0. If
a curve germ α extends to a continuous map α : [0, ε]→ X, we say the extension is the completion
of α, and α is completable. We write α→ x if α has a completion with α(0) = x.
If α : (0, ε] → X has a completion with α → x, then given any neighborhood U of x in X, by
restricting to a smaller interval whenever necessary, we may assume that α([0, ε]) ⊂ U.
We state the following elementary propositions because it will be used several times in this paper.
Proposition 2.2 [2, p. 73]. Let X and Y be semialgebraic sets.
(1) Every curve germ in a compact and semialgebraic set has a completion.
(2) If x belongs to the closure of a semialgebraic subset A of X, then there is a curve germ α in
A with α→ x.
(3) If f : X → Y is semialgebraic and surjective, then every curve germ α in Y lifts to a curve
germ α̃ in X, that is f ◦ α̃ = α.
(4) Every curve germ in X ∪ Y is a curve germ in either X or Y.
Proposition 2.3 [2, p. 73]. Let f : X → Y be a map whose graph is semialgebraic.
(1) f is continuous if and only if for any completable curve germ α in X with α→ x, the curve
germ f ◦ α is also completable in Y with f ◦ α→ f(x).
(2) Suppose f is continuous. Then f is semialgebraically proper if and only if the following
condition holds; if a curve germ α̃ in X such that f ◦ α̃ is completable in Y, then α̃ is completable
in X.
The following lemma is valid if Y is a compactly generated Hausdorff space. Every metric space
is first-countable and therefore compactly generated Hausdorff. Since semialgebraic sets are (usual)
metric spaces, we have the following lemma.
Lemma 2.1. Let f : X → Y be a continuous map between semialgebraic sets. Then the
following are equivalent:
(1) f is proper;
(2) f is closed and its fibers are compact.
The above lemma still valid in the semialgebraic category as in the following proposition. A
semialgebraic map f : X → Y is called semialgebraically closed if f maps every closed and
semialgebraic subset of X to a closed and semialgebraic subset of Y.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1264 DAE HEUI PARK
Proposition 2.4. Let f : X → Y be a semialgebraic map between semialgebraic sets. Then the
following are equivalent:
(1) f is semialgebraically proper;
(2) f is semialgebraically closed and its fibers are compact.
Proof. Let f be a semialgebraically proper. Clearly f−1(y) is compact for all y ∈ Y. Suppose
C is a closed and semialgebraic subset of M and let f(C) denote the closure of f(C) in Y. Given
y ∈ f(C), it follows from Proposition 2.2(2) that there exists a curve germ α in f(C) with α → y.
Since the restriction f |C : C → f(C) is surjective, by Proposition 2.2(3), there is a curve germ α̃
in C such that f ◦ α̃ = α. Thus, since α is completable in Y and f is semialgebraically proper, by
Proposition 2.3(2), α̃ is completable in X. Let α̃ → x, then x ∈ C because C is closed in X. By
Proposition 2.3(1), α = f ◦ α̃→ f(x), and hence y = f(x) ∈ f(C). Therefore f(C) is closed in Y.
Conversely, let f be a semialgebraically closed map such that f−1(y) is compact for all y ∈ Y.
Let K be compact and semialgebraic in Y. By Proposition 2.1(4) f−1(K) is semialgebraic. We
will show that f−1(K) is compact. Let {Uα | α ∈ Λ} be an open cover of f−1(K). For x ∈ Uα
we can take an open and semialgebraic set Wα,x such that x ∈ Wα,x ⊂ Uα. Then the collection
B =
⋃
α∈Λ{Wα,x | x ∈ Uα} is a refinement of {Uα | α ∈ Λ}. It is enough to show that B
contains a finite subcollection that also covers f−1(K). For all z ∈ K, B is also an open cover
of f−1(z). Since the latter is compact, it has a finite subcover. In other words, for each z ∈ K,
there is a finite set Az ⊂ B such that f−1(z) ⊂
⋃
W∈Az
W. The set X −
⋃
W∈Az
W is closed
and semialgebraic in X. Its image is closed in Y, because f is a semialgebraically closed map.
Hence the set Vz = Y − f
(
X −
⋃
W∈Az
W
)
is open in Y. It is easy to check that Vz contains the
point z. Since K ⊂
⋃
z∈K Vz and K is compact, there are finitely many points z1, . . . , zn such that
K ⊂
⋃n
i=1 Vzi . Furthermore the set B∗ =
⋃n
i=1Azi is a finite union of finite sets, thus B∗ is finite.
Since f−1(K) ⊂ f−1 (
⋃n
i=1 Vzi) ⊂
⋃
W∈B∗ W, we have found a finite subcover of f−1(K).
Proposition 2.4 is proved.
Theorem 2.1. Let f : X → Y be a semialgebraic map between semialgebraic sets. Then the
following are equivalent:
(1) f is semialgebraically proper;
(2) f is proper.
Proof. Let f be a semialgebraically proper map. Clearly, all fibers f−1(y), y ∈ Y, are compact.
It suffices to show that f is closed map. Let C be a closed subset of X. Suppose f(C) is not closed
in Y. Then there exist a point y in f(C) which is not contained in f(C). Since f−1(y) is disjoint
from the closed set C, for every point x ∈ f−1(y) has a semialgebraic neighborhood Ux which does
not meet C. Since f−1(y) is compact there exist finitely many points x1, . . . , xn ∈ f−1(y) such that
f−1(y) ⊂ Ux1 ∪ . . . ∪ Uxn .
Then the set B = X − (Ux1 ∪ . . . ∪ Uxn) is closed and semialgebraic in X and contains C. Since f
is semialgebraically closed, the image f(B) is closed in Y. Thus f(C) ⊂ f(B). This contradiction
since y /∈ f(B). Therefore f is closed.
The converse is trivial.
Theorem 2.1 is proved.
Let f : X → Y and g : Z → Y be semialgebraic maps. Then the pullback X ×Y Z = {(x, z) ∈
∈ X × Z | f(x) = g(x)} is closed and semialgebraic in X × Z. The pullback diagram
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
NOTE ON SEMIALGEBRAICALLY PROPER MAPS 1265
X ×Y Z
p1
��
p2
// Z
g
��
X
f
// Y
commutes, where p1 and p2 are the canonical projections.
Theorem 2.2. Let f : X → Y be a semialgebraic map between semialgebraic sets. Then the
following are equivalent:
(1) f is semialgebraically proper;
(2) for every semialgebraic map g : Z → Y, the canonical projection p2 : X ×Y Z → Z is
semialgebraically proper.
Proof. Let f be a semialgebraically proper map. Let g : Z → Y be a semialgebraic map.
Suppose K is compact and semialgebraic in Y. Sice the preimage p−1
2 (K) is a closed subset of a
compact set f−1(g(K))×K, it is semialgebraic and compact. Hence p2 is semialgebraically proper.
Conversely, taking Z = Y and g the identity map on Y, it follows immediately that f is
semialgebraically proper.
Theorem 2.2 is proved.
3. Semialgebraically proper actions. The definition of a semialgebraic group is similar to that
of a Lie group, i.e., a semialgebraic set G is called a semialgebraic group if it is a topological group
such that the group multiplication and the inversion are semialgebraic. Every semialgebraic group
has a Lie group structure, and hence locally compact. Moreover, every semialgebraic subgroup of a
semialgebraic group is closed (see [7, 8]). In this section G always a semialgebraic group.
By a semialgebraic transformation group we mean a triple (G,X, θ), where G is a semialgebraic
group, X is a semialgebraic set, and θ : G×X → X is a semialgebraic map such that
(1) θ(g, θ(h, x)) = θ(gh, x) for all g, h ∈ G and x ∈ X;
(2) θ(e, x) = x for all x ∈ X, where e is the identity of G.
In this case X is called a semialgebraic G-set, and θ is called the semialgebraic action of G on X.
As usual we shortly write gx for θ(g, x). A semialgebraic G-set X is called semialgebraically proper
if the map
ϑ∗ : G×X → X ×X, (g, x) 7→ (gx, x)
is semialgebraically proper. Clearly, if G is compact, then every semialgebraic G-set is semial-
gebraically proper. If G is not compact but X is compact, then X is not semialgebraically proper.
Moreover, if X is a semialgebraically proper G-set, then the orbit space X/G is Hausdorff. Similarly,
a G-space is called proper if the map ϑ∗ is proper.
Theorem 2.1 implies that ϑ∗ is semialgebraically proper if and only if it is proper. Thus we have
the following theorem.
Theorem 3.1. Let X be a semialgebraic G-set. Then the following are equivalent:
(1) X is a semialgebraically proper G-set;
(2) X is a proper G-space.
Proposition 3.1. Let X be a semialgebraic G-set. Then the following are equivalent:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1266 DAE HEUI PARK
(1) X is a semialgebraically proper G-set;
(2) if two curve germs α in X and γ in G such that α and γα are completable, then γ is
completable.
Proof. This follows from Proposition 2.3(2).
Lemma 3.1. Let X be a semialgebraically proper G-set. If H is a semialgebraic subgroup of
G and K is compact and semialgebraic in X, then the set HK = {hx | h ∈ H, x ∈ K} is closed
and semialgebraic in X.
Proof. Clearly HK = θ(H × K). It follows that the set HK is semialgebraic by Proposi-
tion 2.1(4). We now prove that HK is closed in X. For x ∈ HK, there exists a curve germ α in
HK with α → x. Since the restriction θ| : H ×K → HK is surjective, we can find curve germs
γ ⊂ H and β ⊂ K such that γβ = α. By Proposition 2.2, β is completable in K, say β → y.
By Proposition 3.1, γ is completable in G, say γ → g. Since H is closed, g ∈ H, and hence
x = gy ∈ HK.
Lemma 3.1 is proved.
In particular, for each x ∈ X, the orbit G(x) is closed in X.
If the map θ is semialgebraically proper, then the map ϑ∗ is also semialgebraically proper by
Propositions 2.3 and 3.1. But the converse does not hold.
Example 3.1. We consider the semialgebraic group R∗ of nonzero real numbers under multipli-
cation. Let X = R2 − {0}. We can view X as a semialgebraic R∗-set with the action
θ : R∗ ×X → X, θ(t,x) = tx.
Then ϑ∗ is semialgebraically proper, and hence X is semialgebraically proper. Indeed, given two
curve germs α in X and γ in R∗ such that α and γα are completable, say α→ a and γα→ b, then
γ → 〈a,b〉
〈a,a〉
∈ R∗ where 〈, 〉 denotes the inner product, and thus γ is completable. So it follows that
ϑ∗ is semialgebraically proper by Proposition 3.1.
On the other hand, let H = {t ∈ R∗ | t > 0} and C = {(x, y) ∈ X | xy = 1}, then H × C
is closed and semialgebraic in R∗ ×X. Since the image θ(H × C) = {(x, y) ∈ X | xy > 0} is not
closed in X, θ is not semialgebraically closed, and hence not semialgebraically proper.
For two subsets U and V of X, we set
((U, V )) = {g ∈ G | U ∩ gV 6= ∅}.
Proposition 3.2. Let X be a semialgebraic G-set. Then the following are equivalent:
(1) X is a semialgebraically proper G-set;
(2) for any two compact subsets K1 and K2 of M, the subset ((K1,K2)) of G is compact.
Proof. Suppose X is a semialgebraically proper G-set. Then ϑ∗ is semialgebraically proper, and
hence proper. For given two compact subsets K1 and K2 of M, ϑ−1
∗ (K1 ×K2) is a compact subset
of G×X. Let p : G×X → G be the canonical projection. Then ((K1,K2)) = p(ϑ−1
∗ (K1 ×K2)),
and hence ((K1,K2)) is compact.
Conversely, suppose K is compact and semialgebraic in X ×X. Let p1 and p2 be the canonical
projections of X × X onto its first and second factors, respectively. Then K1 = p1(K) and
K2 = p2(K) are compact and semialgebraic in X. Obviously, we see that
ϑ−1
∗ (K) ⊂ ϑ−1
∗ (K1 ×K2) ⊂ ((K1,K2))×K2.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
NOTE ON SEMIALGEBRAICALLY PROPER MAPS 1267
Since ϑ−1
∗ (K) is closed and ((K1,K2))×K2 is compact, ϑ−1
∗ (K) is compact. Hence ϑ∗ is semial-
gebraically proper.
Proposition 3.2 is proved.
In particular, for each x ∈ X, the isotropy subgroup Gx = (({x}, {x})) is compact.
Now we want to compare this notion of proper with that of Palais in [6]. Let X be a G-space. A
subset U of X is called thin if the set ((U,U)) has compact closure in G. A G-space X is called a
Cartan G-space if every point x ∈ X has a thin neighborhood. A subset S of a G-space X is called
small if each point x ∈ X has a neighborhood U such that the set ((S,U)) has compact closure. A
G-space X is called by Palais [6] a “proper” if every point x ∈ X has a small neighborhood. We
need to distinguish this notion from the former. To do this, in this case we call X a Palais-proper
G-space.
Proposition 3.3. Let X be a semialgebraic G-set. If X is a Palais-proper G-space, then it is a
semialgebraically proper G-set.
Proof. Suppose X is a Palais-proper G-space. Let K1 and K2 be two compact subsets of X.
We first prove that ((K1,K2)) is a closed subset of G. If g is a point of the closure of ((K1,K2)) in
G, then there is a sequence gn of points of ((K1,K2)) converges to g. For each positive integer n,
we can choose xn ∈ K2 such that gnxn ∈ K1 because K1 ∩ gnK2 6= ∅. Since K2 is compact, there
exists a subsequence xni of xn which converges to x ∈ K2, so that gnixni → gx. Then gx ∈ K1
because K1 is closed, it follows that g ∈ ((K1,K2)). Therefore, ((K1,K2)) is closed in G.
We now prove that ((K1,K2)) is compact. Since X is Palais-proper G-space, for each (x, y) ∈
∈ X ×X, there is an open neighborhood U∗×V∗ of (x, y) such that ((U∗, V∗)) has compact closure.
By the compactness of K1 × K2, there exists a finite open covering {U1 × V1, . . . , Uk × Vk} of
K1 ×K2 such that ((Ui, Vi)) has compact closure for all i. Since ((K1,K2)) ⊂
⋃k
i=1((Ui, Vi)), the
set ((K1,K2)) has compact closure. Then ((K1,K2)) is compact because it is closed. Hence X is
semialgebraically proper by Proposition 3.2.
Proposition 3.3 is proved.
Theorem 3.2. If X be a semialgebraically proper G-set, then it is a Cartan G-space.
Proof. Suppose X is a semialgebraically proper G-set. Then ϑ∗ is semialgebraically proper,
and hence proper. For every x ∈ X, the isotropy subgroup Gx is compact. Since G is locally
compact, there exists an open semialgebraic neighborhood W of Gx in G whose closure is compact.
By Lemma 2.1, ϑ∗ is closed, so that the image ϑ∗((G − W ) × X) is closed in X × X. Since
(x, x) /∈ ϑ∗((G−W )×X), there exist open neighborhoods U1 and U2 of x such that
U1 × U2 ∩ ϑ∗((G−W )×X) = ∅.
It follows that ((U1, U2)) ⊂W. Indeed, if g ∈ ((U1, U2)), then there exists y ∈ U2 such that gy ∈ U1.
Hence ϑ∗(g, y) = (gy, y) ∈ U1×U2. Since U1×U2 ∩ϑ∗((G−W )×X) = ∅, we have g /∈ G−W,
and so g ∈ W. Hence ((U1, U2)) ⊂ W. Therefore, the intersection U1 ∩ U2 is a thin neighborhood
of x.
Theorem 3.2 is proved.
The converse of the above theorem does not hold.
Example 3.2. Let R∗ denote the semialgebraic group of nonzero real numbers under multiplica-
tion, and let X = R2 − {0}. We consider a semialgebraic R∗-set X with the action
θ : R∗ ×X → X, θ(t, (x, y)) =
(
tx,
1
t
y
)
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1268 DAE HEUI PARK
Then X is a Cartan G-space. Indeed, given (x, y) ∈ X, we set
U = (x− |x|/2, x+ |x|/2)× (y − 1, y + 1) if x 6= 0,
U = (x− 1, x+ 1)× (y − |y|/2, y + |y|/2) if y 6= 0,
then ((U,U)) ⊂
[
1
3
, 3
]
, and hence the set U is a thin neighborhood of (x, y).
But X is not semialgebraically proper. Indeed, let e1 = (1, 0), e2 = (0, 1) ∈ R2, and let Ki
denote the closed ball of radius
1
2
centered at ei in Y for i = 1, 2. Then K1 ×K2 is compact and
semialgebraic in X × X, but ϑ−1
∗ (K1 × K2) is not compact because it contains an unbounded set{(
t,
(
1
t
, 1
)) ∣∣∣∣ t ≥ 2
}
. Hence ϑ∗ is not semialgebraically proper.
Corollary 3.1. Assume thatX is a locally complete semialgebraic set. IfX is a semialgebraically
proper G-set, then it is a Palais-proper G-space.
Proof. Since X is a Cartan G-space, every point x ∈ X has a thin neighborhood W. Note that a
semialgebraic set is locally complete if and only if it is locally compact. Since X is locally compact,
we can take a semialgebraic neighborhood U of x such that the closure U is compact and U ⊂W.
We will show that U is a small neighborhood of x. It is enough to show that, for each y ∈ X,
there is a neighborhood V of y in X such that ((U, V )) has compact closure. In case G(y)∩U = ∅,
put V = M −G(U), then it is an open neighborhood of y by Lemma 3.1. Clearly ((U, V )) = ∅. In
case G(y) ∩ U 6= ∅, gy ∈ U for some g ∈ G. Let V = g−1W, we have ((U, V )) = ((U,W ))g ⊂
⊂ ((W,W ))g. Since ((W,W )) has compact closure, so does ((U, V )).
Corollary 3.1 is proved.
There is an example of a proper action of R on a Gδ-subset X of R2 which is not Palais-proper
(see [5, p. 79]). But, in this case, the R-space X is not semialgebraic.
We conclude this paper with a natural question.
Question. Does there exists a semialgebraically proper G-set which is not Palais-proper?
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Received 10.09.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
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| id | umjimathkievua-article-2217 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:20:56Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/83/9f98134f3daf8a890f96c65270106683.pdf |
| spelling | umjimathkievua-article-22172019-12-05T10:26:31Z A Note on Semialgebraically Proper Maps Про напiвалгебраїчно власні відображення Park, Dae Heui Парк, Де Ху We prove that a semialgebraic map is semialgebraically proper if and only if it is proper. As an application of this assertion, we compare the semialgebraically proper actions with proper actions in a sense of Palais. Доведено, що напівалгебраїчне відображення є напівалгебраїчно власним тоді і тільки тоді, коли воно є власним. Як застосування цього факту, порівнюються напівалгебраїчно власні дії з власними діями в сенсі Пале. Institute of Mathematics, NAS of Ukraine 2014-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2217 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 9 (2014); 1262–1268 Український математичний журнал; Том 66 № 9 (2014); 1262–1268 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2217/1435 https://umj.imath.kiev.ua/index.php/umj/article/view/2217/1436 Copyright (c) 2014 Park Dae Heui |
| spellingShingle | Park, Dae Heui Парк, Де Ху A Note on Semialgebraically Proper Maps |
| title | A Note on Semialgebraically Proper Maps |
| title_alt | Про напiвалгебраїчно власні відображення |
| title_full | A Note on Semialgebraically Proper Maps |
| title_fullStr | A Note on Semialgebraically Proper Maps |
| title_full_unstemmed | A Note on Semialgebraically Proper Maps |
| title_short | A Note on Semialgebraically Proper Maps |
| title_sort | note on semialgebraically proper maps |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2217 |
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