A note on discrete C-embedded subspaces
-
Mehrdad Namdari
and
Mohammad Ali Siavoshi
Abstract
It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of υ X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of υ X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact.
(Communicated by Ľubica Holá)
Acknowledgement
The authors would like to thank professor O. A. S. Karamzadeh for his helpful discussion and encouragement during the preparation of this article. They are also indebted to the referee for carefully reading the article and giving some very useful comments.
References
[1] Azarpanah, F.—Karamzadeh, O. A. S. : Algebraic Characterizations of some disconnected spaces, Ital. J. Pure Appl. Math. 12 (2002), 155–168.Search in Google Scholar
[2] Comfort, W.—Hager, A. : Maximal realcompact (and other) topologies, Topology Proc. 28(2) (2004), 425–443.Search in Google Scholar
[3] Dow, A. : Maximal realcompact spaces and measurable cardinals, Topology Appl. 150 (2005), 1–5.10.1016/j.topol.2004.11.010Search in Google Scholar
[4] Engelking, R. : General Topology, Heldermann Verlag, Berlin, 1989.Search in Google Scholar
[5] Gillman, L.—Jerison, M. : Rings of Continuous Functions, Springer-Verlag, New York, 1976.Search in Google Scholar
[6] Karamzadeh, O. A. S.—Namdari, M.—Siavoshi, M. A. : A note on λ-compact spaces, Math. Slovaca 63(6) (2013), 1371–1380.10.2478/s12175-013-0177-3Search in Google Scholar
[7] Karamzadeh, O. A. S.—Rostami, M. : On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 94(1) (1985), 179–184.10.2307/2044578Search in Google Scholar
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Articles in the same Issue
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
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- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
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