Blumberg theorem

In 1922, H. Blumberg [a1] proved that if $ X = Y = \mathbf R $, the real number set, then:

$$ \tag{a1 } \forall f : X \rightarrow Y \exists D \subset X, \textrm{ dense in } X: $$

$$ f \mid _ {D} \textrm{ continuous } . $$

Even for functions $ f : \mathbf R \rightarrow \mathbf R $, the set $ D $ in (a1) cannot be made to have cardinality $ c $( see [a2]). S. Baldwin (see the references of [a6]) showed that it is consistent with the axioms of set theory that the set $ D $ in (a1) can always be chosen to be uncountably dense; the set $ D $ cannot be necessarily chosen so that, for one-to-one functions, $ f \mid _ {D} $ is a homeomorphism (C. Goffman; see the references of [a6]); further, the set $ D $ cannot necessarily be chosen so as to make $ f \mid _ {D} $ differentiable or monotonic (see J. Cedar; see the references of [a6]). Despite the above, J.B. Brown (see the references of [a6]) proved that for every $ f : \mathbf R \rightarrow \mathbf R $ there exists a set $ N \subset \mathbf R $, $ N $ being $ c $- dense in $ \mathbf R $, such that $ f \mid _ {N} $ is pointwise discontinuous (relative to $ N $).

Blumberg spaces.

Let $ Y = \mathbf R $. A space $ X $ is called a Blumberg space if (a1) holds for $ X $. J.C. Bradford and Goffman [a3] proved that a metric space $ X $ is Blumberg if and only if $ X $ is a Baire space, i.e., a space in which open, non-empty subsets are of the second category. The key lemma in their proof is the Banach category theorem. H.E. White [a4] extended the Bradford–Goffman theorem to topological spaces $ X $ which have $ \sigma $- disjoint pseudo-bases (cf. also Topological space). He also showed that the real number set $ \mathbf R $ with the density topology is a Baire space which is not Blumberg. W.A.R. Weiss (see the references of [a5]) gave an example of a compact Hausdorff space which is not Blumberg. Z. Piotrowski and A. Szymański [a7] showed that if $ X $ is a space for which (a1) holds with $ Y = \mathbf R $, then (a1) holds for every second-countable space $ Y $. The following characterization is of interest (see [a7]): A space is Blumberg if and only if for every countable covering $ {\mathcal P} $ of $ X $ there exists a dense subset $ D $ of $ X $ such that $ P \cap D $ is open in $ D $ for every $ P \in {\mathcal P} $, see also [a8] for more characterizations. General references for this area are [a5], [a6].

The simple example of the identity function from the real number set with the Euclidean topology into the real number set with the discrete topology exhibits the necessity of certain restrictions placed upon the range space. Spaces $ Y $ for which (a1) holds, where $ X = \mathbf R $, were studied in [a9].

The dynamics of Blumberg spaces.

It is easy to see that a dense or closed subspace of a Blumberg space need not be Blumberg. The Stone–Čech compactification of a dense subspace of a completely regular Blumberg space is a Blumberg space, a result of R. Levy and R.H. McDowell (see the references of [a7]).

The Cartesian product of Blumberg spaces need not be a Blumberg space, since there is a metric Baire space (hence Blumberg space) whose square is not Baire. On the other hand, S. Todorčević [a10] showed that there is a first-countable compact space $ X $ that is not Blumberg, whereas $ X \times X $ is a Blumberg space. It follows from the above theorem that the image of a Blumberg space under an open and continuous function need not be Blumberg.

Consider the union $ X $ of the graph $ Z $ of the function $ f $ defined by $ f ( x ) = {1 / q } $ if $ x = {p / q } $ in lowest terms, $ 0 < x <1 $, and a copy $ Y $ of the rational number set between $ 0 $ and $ 1 $. The natural projection of $ X $ onto $ Y $( which is constant on $ Y $) shows that even perfect, continuous functions do not necessarily preserve Blumberg spaces. In contrast, Blumberg spaces are preserved in pre-images under irreducible surjections [a7].

M. Valdivia [a11] showed that (a1) holds for linear transformations, where $ X $ and $ Y $ are metrizable linear spaces and $ X $ is of the second category. L. Drewnowski subsequently proved that "dense subset" in Valdivia's theorem cannot be replaced by "dense linear subspace" .

Blumberg sets, i.e. dense sets $ D $ appearing in (a1), have been studied in connection with characterizations of certain almost-continuous functions, such as quasi-continuous functions ([a13]).

References

[a1]	H. Blumberg, "New properties of all real functions" Trans. Amer. Math. Soc. , 3 (1922) pp. 113–128
[a2]	W. Sierpiński, A. Zygmund, "Sur une fonction qui est discontinue sur tout ensemble de puissance de continue" Fundam. Math. , 4 (1923) pp. 316–318
[a3]	J.C. Bradford, C. Goffman, "Metric spaces in which Blumberg's theorem holds" Proc. Amer. Math. Soc. , 11 (1960) pp. 667–670
[a4]	H.E. White, Jr., "Topological spaces in which Blumberg's theorem holds" Proc. Amer. Math. Soc. , 44 (1974) pp. 454–462
[a5]	J.B. Brown, "Variations on Blumberg's theorem" Real Anal. Exchange , 9 (1983/84) pp. 123–137
[a6]	J.B. Brown, "Restriction theorems in real analysis" Real Anal. Exchange , 20 (1994/5) pp. 510–526
[a7]	Z. Piotrowski, A. Szymanski, "Concerning Blumberg's theorem" Houston J. Math. , 10 (1984) pp. 109–115
[a8]	A. Szymański, "On $m$-Baire and $m$-Blumberg spaces" , Proc. Conf. Topology and Measure II (Rostock–Warnemunde, GDR, 1977) , Part I , Greifswald (1980) pp. 151–161
[a9]	J.B. Brown, Z. Piotrowski, "Co-Blumberg spaces" Proc. Amer. Math. Soc. , 96 (1986) pp. 686–688
[a10]	S. Todorčević, "Stationary sets, trees and continuums" Publ. Inst. Math. , 27 (1981) pp. 249–262
[a11]	M. Valdivia, "On the closed graph theorem in topological spaces" Manuscr. Math. , 23 (1978) pp. 173–184
[a12]	M. Wilhelm, "Nearly lower semicontinuity and its applications" , General Topology and its Applications (Fifth Prague Topology Symp.) , Heldermann (1981) pp. 692–698
[a13]	T. Nebrunn, "Quasi-continuity" Real Anal. Exchange , 14 (1988–89) pp. 259–306