^{1}Department of Mathematics, Chamran University, Ahwaz, Iran.

^{2}Department of Mathematics, Persian Gulf University, Boushehr, Iran.

Abstract

A topological space is called submaximal if each of its dense subsets is open and is called nodec if each of its nowhere dense ea subsets is closed. Here, we study a variety of spaces some of which have already been studied in $C(X)$. Among them are, most importantly, quasi $P$-spaces and pointwise quasi $P$-spaces. We obtain some new useful topological characterizations of quasi $P$-spaces and pointwise quasi $P$-spaces. Consequently, we obtain a close relation between these latter spaces and submaximal and nodec spaces.

E. Abu Osba and M. Henriksen, Essential P-spaces: a generalization of door spaces, Comment. Math. Univ. Corolin. 45 (2004), no. 3, 509--518.

A.R. Aliabad, z◦-ideals in C(X), PhD Thesis, (in Persian), 1996.

A.R. Aliabad, Tree topology, Int. J. Contemp. Math. Sci. 5 (2010), no. 21, 1045--1054.

A.R. Aliabad and M. Badie, Connections between C(X) and C(Y ), where Y is a subspace of X, Bull. Iranian Math. Soc. 37 (2011), no. 4, 109--126.

A.V. Arhangel'ski_i and P.J. Collins, On submaximal spaces, Topology Appl. 64 (1995), no. 3, 219--241.

F. Azarpanah, O.A.S. Karamzadeh and A. Rezaei Aliabad, On ideals consisting entirely of zero divisors, Comm. Algebra 28 (2000), no. 2, 1061--1073.

F. Azarpanah and M. Karavan, On nonregular ideals and zo-ideals in C(X), Czechoslo-vak Math. J. 55 (130) (2005), no. 2, 397--407.

F. Azarpanah and R. Mohamadian, pz-ideals and pz◦-ideals in C(X), Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 6, 989--996.

N. Bourbaki, Topologic General, Actualites Scientiques et Industrielles 1142, Hermann, 3rd edition, Paris, 1961.

E.K. van Douwen, Application of maximal topologies, Topology Appl. 51 (1993), no. 2, 125--139.

L. Gillman and M. Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc. 77 (1954) 340--362.

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, 1976.

E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943) 309--333.

F. Hernandez-Hernandez, O. Pavlov, P.J. Szeptycki and A.H. Tomita, Realcompactness in maximal and submaximal spaces, Topology Appl. 154 (2007), no. 16, 2997--3004.

M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 110 (1965) 110--130.

M. Henriksen, J. Martinez and R.G. Woods, Spaces X in which all prime z-ideals of C(X) are minimal or maximal, Comment. Math. Univ. Carolin. 44 (2003), no. 2, 261--294.

M. Henriksen and R.G.Woods, Cozero complemented spaces; when the space of minimal prime ideals of C(X) is compact, Topology Appl. 141 (2004), no. 1-3, 147--170.

M. Karavan, On QP-spaces, in: International Conference of Mathematics, Beijing China, Aug. 2002.

R. Levy, Almost P-spaces, Canad. J. Math. 29 (1977) 284--288.

J. Martinez and E.R. Zenk, Dimension in algebraic frames, II: applications to frames of ideals in C(X), Comment. Math. Univ. Carolin. 46 (2005), no. 4, 607--636.

J. Martinez, The role of frames in the development of lattice-ordered groups: a personal account, Positivity, 161--195, Trends Math., Birkhäuser, Basel, 2007.

T. Terada, On remote points in _X n X, Proc. Amer. Math. Soc. 77 (1979), no. 2, 264--266.

A.I. Veksler, P′-points, P′-sets, P′-spaces, A new class of order-continuous measures and functions, Soviet Math. Dokl. 14 (1973) 1445-1450.

M. Weir, Hewitt-Nachbin Spaces, North Holland Publ. Co. Amsterdam, 1975.