View source for Blumberg theorem

In 1922, H. Blumberg [[#References|[a1]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106501.png" />, the real number set, then:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106503.png" /></td> </tr></table>

Even for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106504.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106505.png" /> in (a1) cannot be made to have cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106506.png" /> (see [[#References|[a2]]]). S. Baldwin (see the references of [[#References|[a6]]]) showed that it is consistent with the axioms of set theory that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106507.png" /> in (a1) can always be chosen to be uncountably dense; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106508.png" /> cannot be necessarily chosen so that, for one-to-one functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b1106509.png" /> is a [[Homeomorphism|homeomorphism]] (C. Goffman; see the references of [[#References|[a6]]]); further, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065010.png" /> cannot necessarily be chosen so as to make <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065011.png" /> differentiable or monotonic (see J. Cedar; see the references of [[#References|[a6]]]). Despite the above, J.B. Brown (see the references of [[#References|[a6]]]) proved that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065012.png" /> there exists a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065014.png" /> being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065015.png" />-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065016.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065017.png" /> is pointwise discontinuous (relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065018.png" />).

==Blumberg spaces.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065019.png" />. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065020.png" /> is called a Blumberg space if (a1) holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065021.png" />. J.C. Bradford and Goffman [[#References|[a3]]] proved that a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065022.png" /> is Blumberg if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065023.png" /> is a [[Baire space|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 [[#References|[a4]]] extended the Bradford–Goffman theorem to topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065024.png" /> which have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065025.png" />-disjoint pseudo-bases (cf. also [[Topological space|Topological space]]). He also showed that the real number set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065026.png" /> with the density topology is a Baire space which is not Blumberg. W.A.R. Weiss (see the references of [[#References|[a5]]]) gave an example of a compact [[Hausdorff space|Hausdorff space]] which is not Blumberg. Z. Piotrowski and A. Szymański [[#References|[a7]]] showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065027.png" /> is a space for which (a1) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065028.png" />, then (a1) holds for every second-countable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065029.png" />. The following characterization is of interest (see [[#References|[a7]]]): A space is Blumberg if and only if for every countable covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065031.png" /> there exists a dense subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065034.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065035.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065036.png" />, see also [[#References|[a8]]] for more characterizations. General references for this area are [[#References|[a5]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065037.png" /> for which (a1) holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065038.png" />, were studied in [[#References|[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|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 [[#References|[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 [[#References|[a10]]] showed that there is a first-countable compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065039.png" /> that is not Blumberg, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065041.png" /> of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065042.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065043.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065044.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065045.png" /> in lowest terms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065046.png" />, and a copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065047.png" /> of the rational number set between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065049.png" />. The natural projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065050.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065051.png" /> (which is constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065052.png" />) shows that even perfect, continuous functions do not necessarily preserve Blumberg spaces. In contrast, Blumberg spaces are preserved in pre-images under irreducible surjections [[#References|[a7]]].

M. Valdivia [[#References|[a11]]] showed that (a1) holds for linear transformations, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065054.png" /> are metrizable linear spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065055.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065056.png" /> appearing in (a1), have been studied in connection with characterizations of certain almost-continuous functions, such as quasi-continuous functions ([[#References|[a13]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Blumberg,   "New properties of all real functions"  ''Trans. Amer. Math. Soc.'' , '''3'''  (1922)  pp. 113–128</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Sierpiński,   A. Zygmund,   "Sur une fonction que est discontinue sur tout ensemble de puissance de continue"  ''Fundam. Math.'' , '''4'''  (1923)  pp. 316–318</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.C. Bradford,   C. Goffman,   "Metric spaces in which Blumberg's theorem holds"  ''Proc. Amer. Math. Soc.'' , '''11'''  (1960)  pp. 667–670</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.E. White, Jr.,   "Topological spaces in which Blumberg's theorem holds"  ''Proc. Amer. Math. Soc.'' , '''44'''  (1974)  pp. 454–462</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.B. Brown,   "Variations on Blumberg's theorem"  ''Real Anal. Exchange'' , '''9'''  (1983/84)  pp. 123–137</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.B. Brown,   "Restriction theorems in real analysis"  ''Real Anal. Exchange'' , '''20'''  (1994/5)  pp. 510–526</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Z. Piotrowski,   A. Szymanski,   "Concerning Blumberg's theorem"  ''Houston J. Math.'' , '''10'''  (1984)  pp. 109–115</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Szymański,   "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065057.png" />-Baire and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110650/b11065058.png" />-Blumberg spaces" , ''Proc. Conf. Topology and Measure II (Rostock–Warnemunde, GDR, 1977)'' , '''Part I''' , Greifswald  (1980)  pp. 151–161</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.B. Brown,   Z. Piotrowski,   "Co-Blumberg spaces"  ''Proc. Amer. Math. Soc.'' , '''96'''  (1986)  pp. 686–688</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S. Todorčević,   "Stationary sets, trees and continuums"  ''Publ. Inst. Math.'' , '''27'''  (1981)  pp. 249–262</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M. Valdivia,   "On the closed graph theorem in topological spaces"  ''Manuscr. Math.'' , '''23'''  (1978)  pp. 173–184</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Wilhelm,   "Nearly lower semicontinuity and its applications" , ''General Topology and its Applications (Fifth Prague Topology Symp.)'' , Heldermann  (1981)  pp. 692–698</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  T. Nebrunn,   "Quasi-continuity"  ''Real Anal. Exchange'' , '''14'''  (1988–89)  pp. 259–306</TD></TR></table>