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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