Description:
A proof that Form 304 (There does not exist a
\(T_{2}\) topological space \(X\) such that every infinite subset of \(X\)
contains an infinite compact subset.) is true in \(\cal N1\).
Content:
Form 304 (There does not exist a
\(T_{2}\) topological space \(X\) such that every infinite subset of \(X\)
contains an infinite compact subset.) is true in \(\cal N1\).
Let \(X\) be an infinite T\(_2\) topological space in \(\cal N1\). We need to
show that there is an infinite \(Y\subseteq X\) (\(Y\in\) \(\cal N1\)) such that \(Y\)
has no infinite compact subsets (in \(\cal N1\)). If \(X\) is well orderable then it is
clear that such a \(Y\) exists.Assume that \(X\) is not well orderable in \(\cal N1\).
Choose an \(x\in X\), a support \(E\), an element \(a\in A\), and a permutation \(\phi\in G\)
which fixes \(E\) pointwise so that
(a)\(E\) is a support of \(X\) and its topology.
(b)\(E \cup \{a\}\) is a support of \(x\) and
(c)\(\phi(x) \ne x\). (And therefore by (b) \(\phi(a)\ne a\).)
For all \(\psi,\eta \in G\) such that \(\psi\) and \(\eta\) fix \(E\) pointwise, \(\psi(x) = \eta(x)\) if and only if
\(\psi(a) = \eta(a)\).
If \(\psi(a) = \eta(a)\), then \(\psi\) and \(\eta\) agree on a support of \(x\) and therefore \(\psi(x) = \eta(x)\).
Conversely, if \(\psi(a) \ne \eta(a)\), then the permutation \(\beta\), defined to be the product of the two transpositions
\(\beta = (a,\psi(a))(\phi(a),\eta(a)) \in G\), fixes \(E\) pointwise, agrees with \(\psi\) on a support of \(x\), and
agrees with \(\eta\phi^{-1}\) on a support of \(\phi(x)\). It follows that
\(\beta(x) = \psi(x)\) and that \(\beta(\phi(x)) = \eta\phi^{-1}(\phi(x)) =\eta(x)\). Since \(x \ne \phi(x)\) we conclude that
\(\beta(x) \ne\beta(\phi(x))\), that is \()\psi(x) \ne \eta(x)\).
By the lemma, the set of pairs \(F = \{(\psi(x),\psi(a)): \psi\in G\) and
\(\psi\) fixes \(E\) pointwise\(\}\) is a one to one function (in the model \(\cal N1\)) from
\(Y =_{\hbox{def}}\{\psi(x): \psi\in G\) and \(\psi\) fixes \(E\) pointwise\(\}\) onto \(A - E\).
Since the topology on \(X\) is \(T_{2}\) we can choose open sets
\(C\) and \(D\) so that \(x\in C\), \(\phi(x)\in D\) and \(C\cap D = \emptyset\). Since \(Y\) can be put in a one to one
correspondence with a subset of the atoms in the model, every subset of \(Y\) in the model must be finite or cofinite.
This means that at least one of \(Y\cap C\) or \(Y\cap D\) is finite. We may assume that \(Y\cap C\) is finite by replacing
\(C\) by \(C \cap \phi^{-1}(D)\) if necessary. Under this assumption we can conclude that \(\cal C = \{\psi(C): \psi\in G\) and
\(\psi\) fixes \(E\) pointwise \(\}\) is an opencover for \(Y\). Further each element of \(\cal C\) is finite. It follows that for
any infinite subset \(Z\) of \(Y\), \(\cal C\) is an open cover for \(Z\) without a finite subcover.
Howard-Rubin number:
116
Type:
proof of result
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