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Transcript
PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 51, Number 2, Septembet
1975
THE REGULAR CONTINUOUSIMAGEOF A MINIMALREGULAR
SPACE IS NOT NECESSARILY MINIMALREGULAR
MANUEL P. BERRIOZABAI (BERRI) AND CARROLL F. BLAKEMORE
ABSTRACT.
Herrlich
regular-closed
to show
space
that
for minimal
that
Herrlich's
regular
result
a continuous
pact
that
tinuous
image
of a regular-closed
we give
an example
to a regular
space
Definition
filter-base
regular.
spaces
Example
I.
lar continuous
minimal
This
image
regular
ular-closed,
but not minimal
Presented
spaces
is com-
space
con-
In the following,
regular
continuous
which
shows
that a
regular-closed)
on a topological
exists
space
V £J
characterizations
of regular
of a minimal
regular.
space
in [2] and let
that
V Ç U.
of regular-closed
filter-bases.
regular
Let
is an open
such
space
whose
regu-
(Z, r) be the noncompact
(T, o) be the subspace
of
e/VÎ. In [4] it is shown that (T, o) is reg-
regular
January
whose
an example
in terms
since
! is a regular
to the Society,
of compact
space
[4] that the regular
(and hence
working
constructed
spaces
from a compact
shown
U £ J" there
is an example
p £ G, T - G is compact
space
closed.
is not minimal
space
give
regular
(Z, r) given by T = ipiu UiZj£
properties
of a compact
space
filter-base
standard
regular
a regular
is regular-closed.
regular
We also
A regular
and minimal
result
shows
and compact
function
has
space
that for each
[2] and [3] contain
image
Herrlich
is not necessarily
such
onto
spaces
any continuous
from a minimal
[3].
J
all regular
of a minimal
function
example
space
of a
is given
to a corresponding
of this
regular
continuous
is closed.
is not minimal
be extended
image
an example
Two of the fundamental
the Hausdorff
space
image
continuous
paper,
function.
discussion,
and its consequence
regular
a modification
from a minimal
a closed
to a Hausdorff
continuous
the
In this
cannot
to be Hausdorff.
are that
that
Also,
function
In the subsequent
spaces
shown
spaces.
is not necessarily
are assumed
has
is regular-closed.
the filter-base
filter-base
23,
1975;
y = ÍG Ç T|G
on T with a unique
received
by the editors
June
£ o,
adherent
12,
1974.
AMS (MOS) subject
classifications
Key words and phrases.
continuous
functions.
Minimal
(1970).
regular
Primary 54D25; Secondary 54D30.
spaces,
regular-closed
spaces,
closed
Copyright © 1975, American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
453
454
M. P. BERRIOZABAL (BERRI) AND C. F. BLAKEMORE
point
p which
does
not converge
to p on
T.
Now define
tion / from Z onto T:
if
if z £ T,
fiz) =/p,P,
if z = q,
Íz,
the function
(Z, t) about
the Z .-sheet
from Z onto
regular
T.
This
Example
2.
space
This
exists
a strictly
ty mapping
half of the space
T, and
/ is a continuous
example
since
of (T, 0") onto
where
function
from the minimal
i is not closed,
closed in iT,o').
closed
function
(Z, r) is minimal
solve
a minimal
regular
Clearly
o
space
on T.
Let
/ is continuous,
in Example
exists
from a minimal
function.
Let
(T, ff) is not minimal
topology
1.
Then
set
regular,
but not closed.
g is a continuous
space
F in (T, a) which
in ÍT, o ).
(Z, r)
i be the identi-
(Z, r) onto the regular
a closed
F is not closed
(T, o ).
is not
= F. But /_1(F)
Hence,
is
g is not a closed
our example.
this paper
problem
regular
(T, o- ).
/ is given
there
and we have
We conclude
1. Since
regular
function
is not a closed
Then gif~ lÍF)) = (/ ° /)(/" \f))
in (Z, r) and
function
of a continuous
which
in Example
g = i°/
Since
space
weaker
Now let
from
the desired
is an example
onto a regular
and (T, o) be as given
would
the nonpositive
and (T, o) is not.
regular
there
/ "reflects"
onto the space
yields
func-
if z = in, x, y) and n < 0.
(-72 + 1, x, y),
Geometrically,
the following
with a question
14 in [3] proposed
space
having
(X, r) onto a regular
to which
by Banaschewski
the property
space
an affirmative
that
is closed,
then
every
in [l],
continuous
answer
// (X, r) is
function
is (X, r) compact?
REFERENCES
..
1.
B. Banaschewski,
Über
zwei
Math. Nachr. 13 (1955), 141-150.
2.
M. P.
Berri
and R. H. Sorgenfrey,
Soc. 14 (1963), 454-458.
3. M. P. Berri,
topological
spaces,
Extremaleigenschaften
topologischen
Räume,
MR 17, 66.
Minimal
regular
spaces,
Proc.
Amer.
Math.
MR 27 #2949.
J. R. Porter
and R. M. Stephenson,
General
Topology
and Its Relations
gebra, Proc. Kanpur Topological
Conference,
4. H. Herrlich, T^Abgeschlossenheit
285-294. MR 32 #1664.
Jr., A survey of minimal
to Modern Analysis
and Al-
1968, pp. 93—114.
MR 43 #3895.
und T -MinimalitSt, Math. Z. 88 (1965),
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEW ORLEANS, NEW ORLEANS,
LOUISIANA 70122
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use