Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 05 Aug 2021 02:19:17 GMT2021-08-05T02:19:17Z50111- On regular and homological closure operatorshttp://hdl.handle.net/10316/43635Title: On regular and homological closure operators
Authors: Clementino, Maria Manuel; Gutierres, Gonçalo
Abstract: Observing that weak heredity of regular closure operators in Top and of homological closure operators in homological categories identifies torsion theories, we study these closure operators in parallel, showing that regular closure operators play the same role in topology as homological closure operators do algebraically.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/436352010-01-01T00:00:00Z
- Sequential topological conditions in R in the absence of the axiom of choicehttp://hdl.handle.net/10316/8227Title: Sequential topological conditions in R in the absence of the axiom of choice
Authors: Gutierres, Gonçalo
Abstract: It is known that - assuming the axiom of choice - for subsets A of R the following hold: (a) A is compact iff it is sequentially compact, (b) A is complete iff it is closed in R, (c) R is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/82272003-01-01T00:00:00Z
- On coregular closure operatorshttp://hdl.handle.net/10316/11480Title: On coregular closure operators
Authors: Gutierres, Gonçalo
Abstract: Among closure operators in the sense of Dikranjan and Giuli
[5] the regular
ones have a relevant role and have been widely investigated. On the contrary, the
coregular closure operators were introduced only recently in [3] and they need to
be further investigated. In this paper we study co
regular closure operators, in
connection connectednesses and disconnectednesses, in the realm of topological
spaces and modules.
Fri, 01 Jan 1999 00:00:00 GMThttp://hdl.handle.net/10316/114801999-01-01T00:00:00Z
- What is a first countable space?http://hdl.handle.net/10316/4609Title: What is a first countable space?
Authors: Gutierres, Gonçalo
Abstract: The definition of first countable space is standard and its meaning is very clear. But is that the case in the absence of the Axiom of Choice? The answer is negative because there are at least three choice-free versions of first countability. And, most likely, the usual definition does not correspond to what we want to be a first countable space. The three definitions as well as other characterizations of first countability are presented and it is discussed under which set-theoretic conditions they remain equivalent.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/46092006-01-01T00:00:00Z
- On first and second countable spaces and the axiom of choicehttp://hdl.handle.net/10316/4635Title: On first and second countable spaces and the axiom of choice
Authors: Gutierres, Gonçalo
Abstract: In this paper it is studied the role of the axiom of choice in some theorems in which the concepts of first and second countability are used. Results such as the following are established:
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/10316/46352004-01-01T00:00:00Z
- On regular and homological closure operatorshttp://hdl.handle.net/10316/13629Title: On regular and homological closure operators
Authors: Clementino, Maria Manuel; Gutierres, Gonçalo
Abstract: Observing that weak heredity of regular closure operators in Top and
of homological closure operators in homological categories identifies torsion theories,
we study these closure operators in parallel, showing that regular closure operators
play the same role in topology as homological closure operators do algebraically.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/136292009-01-01T00:00:00Z
- On preLindelöf metric spaces and the Axiom of Choicehttp://hdl.handle.net/10316/13652Title: On preLindelöf metric spaces and the Axiom of Choice
Authors: Gutierres, Gonçalo
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/136522010-01-01T00:00:00Z
- Axioms for sequential convergencehttp://hdl.handle.net/10316/11385Title: Axioms for sequential convergence
Authors: Gutierres, Gonçalo; Hofmann, Dirk
Abstract: It is of general knowledge that those (ultra)filter convergence relations
coming from a topology can be characterized by two natural axioms. However, the
situation changes considerable when moving to sequential spaces. In case of unique
limit points J. Kisynski [Kis60] obtained a result for sequential convergence similar
to the one for ultrafilters, but the general case seems more difficult to deal with.
Finally, the problem was solved by V. Koutnik [Kou85].
In this paper we present an alternative approach to this problem. Our goal is
to find a characterization more related to the case of ultrafilter convergence. We
extend then the result to characterize sequential convergence relations corresponding
to Fréchet topologies, as well to those corresponding to pretopological spaces.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/10316/113852005-01-01T00:00:00Z
- On countable choice and sequential spaceshttp://hdl.handle.net/10316/8212Title: On countable choice and sequential spaces
Authors: Gutierres, Gonçalo
Abstract: Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even R may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of R, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in R if and only if the axiom of countable choice holds for families of subsets of R, and every metric space has a unique -completion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/82122008-01-01T00:00:00Z
- The Ultrafilter Closure in ZFhttp://hdl.handle.net/10316/11238Title: The Ultrafilter Closure in ZF
Authors: Gutierres, Gonçalo
Abstract: It is well known that, in a topological space, the open sets can be
characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without
the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the
ultrafilter convergence determines the open sets for every topological space if and
only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter
Theorem is equivalent to the fact that uX = kX for every topological space X,
where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure
with
uX(A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈ U]}.
However, it is possible to built a topological space X for which uX 6= kX, but the
open sets are characterized by the ultrafilter convergence. To do so, it is proved
that if every set has a free ultrafilter then the Axiom of Countable Choice holds for
families of non-empty finite sets.
It is also investigated under which set theoretic conditions the equality u = k is
true in some subclasses of topological spaces, such as metric spaces, second countable
T0-spaces or {R}.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112382008-01-01T00:00:00Z
- Total Boundedness and the Axiom of Choicehttp://hdl.handle.net/10316/44396Title: Total Boundedness and the Axiom of Choice
Authors: Gutierres, Gonçalo
Abstract: A metric space is Totally Bounded (also called preCompact) if it has a finite ε-net for every ε > 0 and it is preLindelöf if it has a countable ε-net for every ε > 0. Using the Axiom of Countable Choice (CC), one can prove that a metric space is topologically equivalent to a Totally Bounded metric space if and only if it is a preLindelöf space if and only if it is a Lindelöf space. In the absence of CC, it is not clear anymore what should the definition of preLindelöfness be. There are two distinguished options. One says that a metric space X is: (a) preLindelöf if, for every ε > 0, there is a countable cover of X by open balls of radius ?? (Keremedis, Math. Log. Quart. 49, 179–186 2003); (b) Quasi Totally Bounded if, for every ε > 0, there is a countable subset A of X such that the open balls with centers in A and radius ε cover X.
As we will see these two notions are distinct and both can be seen as a good generalization of Total Boundedness. In this paper we investigate the choice-free relations between the classes of preLindelöf spaces and Quasi Totally Bounded spaces, and other related classes, namely the Lindelöf spaces. Although it follows directly from the definitions that every pseudometric Lindelöf space is preLindelöf, the same is not true for Quasi Totally Bounded spaces. Generalizing results and techniques used by Horst Herrlich in [8], it is proven that every pseudometric Lindelöf space is Quasi Totally Bounded iff Countable Choice holds in general or fails even for families of subsets of R
(Theorem 3.5).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/443962016-01-01T00:00:00Z