Luzin space

An uncountable topological $T_1$-space without isolated points in which every nowhere-dense subset is countable. The existence of a Luzin space on the real line follows from the continuum hypothesis. From the negation of the continuum hypothesis and Martin's axiom (cf. Suslin hypothesis) together it follows that no Luzin space exists. In particular, this is compatible with the Zermelo–Fraenkel system of axioms of set theory and the axiom of choice. The existence of metrizable Luzin spaces has been proved under very general assumptions about the place of the cardinality of the continuum in the scale of alephs. Any Luzin space $X$ that lies in a separable metric space $Y$ has the following property: For any sequence $\{\lambda_n\}$ of positive numbers there is a sequence of sets $\{A_n\}$ such that $X=\bigcup_{n=1}^\infty A_n$ and $\delta(A_n)<\lambda_n$, where $\delta(A)$ is the diameter of the set $A$. This property is invariant under continuous mappings. Any continuous image of a Luzin space lying in $Y$ has Lebesgue measure zero and dimension zero. Moreover, it is totally imperfect, that is, it does not contain a Cantor set. The continuum hypothesis implies that there is a regular hereditarily-separable, hereditarily Lindelöf, extremally-disconnected Luzin space of countable $\pi$-weight and with the cardinality of the continuum.

References

[1]	N.N. [N.N. Luzin] Lusin, "Sur un problème de M. Baire" C.R. Acad. Sci. Paris , 158 (1914) pp. 1258–1261
[2]	K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)

Comments

Three slightly different definitions of Luzin space are still in use (apart from whether they must be $T_1$, $T_2$ or $T_3$): An uncountable space all of whose nowhere-dense sets are countable, with 1) no isolated points; or 2) at most $\omega$ isolated points; or 3) any number of isolated points.

References

[a1]	K. Kunen, "Luzin spaces" Topology Proceedings , 1 (1977) pp. 191–199
[a2]	J. Roitman, "Basic $S$ and $L$" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 295–326
[a3]	W. Weiss, "Versions of Martin's axiom" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 827–886
[a4]	A.W. Miller, "Special subsets of the real line" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 201–233