A subset of a topological space is called delta-open if it is the intersection of
countably many open sets.
D. B. Gauld and others defined a topology on the set X=[0, +infinity[ by
using the sets A\cap [x, +infinity[ , where x is in X and A is a co-finite subset
of X as the basic open sets.
They stated that ever dense, delta-open subset of the space X is of the form
A cap [x, +infinity[ , where x is in X and A is a co-countable subset of X.
However, I have not been able to prove this statement, despite that they
considered it to be obvious. Therefore, I ask you to provide me a
reasonable proof.
Their statement is important. Namely, it can be used to prove that
Volterra spaces are genuine generalizations of Baire spaces.
Relevant references are:
1. D. Gauld et al, On Volterra spaces II, Ann. N. Y. Acad. Sci.
806 (1996), 169--173. ( Example 1.)
2. F. A. Ballone, Volterra Spaces, Dissertation, Youngstown State University,
2010, 74 pp. ( Example 5.3.1.)
Dear Peter and Giovanni,
Thank you very much for your kind helps.
Giovanni's observations are already in precise,
publicable forms.
I am in a great trouble, not only with examples,
but also theorems on Volterra spaces.
The literature on Volterra spaces, except the
dissertation of Ballone, is almost unreadable for me.
Your observations indicate that Gauld's Example 3.2
may also need a substantial revision.
Of course, I am working in relator spaces instead the
topological ones.
Some relevant things can be found in Salih's dissertation:
Generalized Open Sets, Minimality and Connectedness
Properties in Relator spaces.
Concerning my recent ideas and problems, I wanted to uppload
an extract from a planned joint book.
However, the System has blocked it with the sentence that :
"The publisher, name unkown".
With best regards,
Arpad
Arpad Szaz I cannot find Example 3.2.
Are you referring to Example 2 or to Example 3?
Dear Giovanni Dore,
This is in the third paper of Gauld et al.
However, for everything it is better to see
Ballone's dissertation.
It is freely available on the Google.
However, for your convenience, I shall now try to
upload it.
Please write me your email address in order that
I could send you everything directly.
ResearchGate has also started to make some
inconveniences.
With best regards,
Arpad Szaz
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