## NONSTANDARD ANALYSIS IN POINT-SET TOPOLOGY

by
Sergio Salbany
,
Todor Todorov

by
Sergio Salbany
,
Todor Todorov

@MISC{Salbany_nonstandardanalysis,

author = {Sergio Salbany and Todor Todorov},

title = {NONSTANDARD ANALYSIS IN POINT-SET TOPOLOGY},

year = {}

}

Abstract We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.

43 |
Applied Nonstandard Analysis
- Davis
- 1977
(Show Context)
Citation Context ...tively). The above statement might be true or false depending on the choice of "; x0 and f . For a more detailed exposition of the formal language L(V (S)) associated with V (S) we refer to (M. Davis =-=[1]-=-, Chapter 1) and (T. Lindstro/m [13], Chapter IV), but we believe that the reader can successfully proceed further without a special background in mathematical logic. After these preparations of the s... |

12 | Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions, in “Nonstandard Analysis: Theory and Practice
- Henson
- 1997
(Show Context)
Citation Context ...e exist also other axiomatic formulations of nonstandard analysis, e.g. H. J. Keisler [11] axiomatization of \Lambda R, the "Internal Set Theory", due to E. Nelson[16] and, more recently, C.W. Henson =-=[7]-=- axiomatic approach. For a discussion and a general overlook we refer again to Tom Lindstro/m [13]. 3. Existence of Nonstandard Models The content of this section can be viewed either as a proof of th... |

11 |
Extended Topology
- Hammer
- 1964
(Show Context)
Citation Context ...\Lambda X, then: (i) A ` _(A). (ii) A ` B implies _(A) ` _(B). (iii) _(_(A)) = _(A). The above lemma shows that the monad of a set is a generalized closure operator in \Lambda X (see e.g. P.C. Hammer =-=[6]-=- and K.D. Stroyan [21], Section 2). (1.6) Corollary: For any A ` \Lambda X and any ff; fi 2 \Lambda X: (i) ff 2 A implies _(ff) ` _(A). (ii) ff 2 _(fi) iff _(ff) ` _(fi). (iii) ff 2 _(fi) and fi 2 _(f... |

4 | Enlargements containing various kinds of completions - Gonshor - 1974 |

4 |
Comments on Nonstandard Topology
- Haddad
- 1978
(Show Context)
Citation Context ...ogy. We shall use as well the terminology of (J.L. Kelley [12]) and (L. Gillman and M. Jerison [3]). For the connection between the standard and nonstandard methods in topology we refer to (L. Haddad =-=[5]-=-). We denote by N and R the sets of the natural and real numbers, respectively, and we also use the notation N0 = f0g [ N. By C(X; R) and Cb(X; R) we shall denote the class of all "continuous" and "co... |

3 |
Non-Standard Characterization of Ideals in
- Dyre
- 1982
(Show Context)
Citation Context ...Then, we have f \Gamma c 2 ker ss so, \Lambda f (ff) = c = ss(f ). Since c is a real number, ff 2 eX . The proof is complete. N Note: The result of the above lemma is related to results in (J.C. Dyre =-=[2]-=-, Theorem (3.3)). The difference with Dyre's work consists in our restriction to real maximal ideals of C(X; R) only and, hence, the localization of ff ine X which is essential for our discussion. (6.... |

3 |
Topologische Reflexionen und
- Herrlich
(Show Context)
Citation Context ...x 2 X. We also have the special property q\Gamma 1[q[G]] = G for all G 2 T , so that q is an open mapping. The space ( eX; eT ) is called the T0 - reflection of (X; T ) (see, for example, H. Herrlich =-=[8]-=-). (3.1) Theorem: Let (X; T ) be a topological space. Then, (X; T ) is weakly Hausdorff iff ( eX ; eT ) is Hausdorff. Proof: Suppose eX is Hausdorff. To show that X is weakly Hausdorff, assume that _(... |

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