Forewords

There are 75 open problems recorded in the Polish journal Fundamenta Mathematicae in volumes 1-34 from its inception, some of which have already been solved. These problems represent the development of the foundations of mathematics since the late 19th century when M. G. Cantor established naive set theory. The original text is in French and German, and the translated version is as follows.

Problems

When is a set of points $P$ a one-to-one and continuous image (but not necessarily bi-continuous) of a set $Q$ and when is $Q$ a one-to-one and continuous image of $P$? Are the sets $P$ and $Q $ necessarily homeomorphic?

Problem of M. W. Sierpinski.

Is a (bounded) planar continuum, topologically homogeneous, necessarily homeomorphic to a circumference?

(A set $E$ is said to be topologically homogeneous, when there exists for any pair of points $a, b$ of $E$ a one-to-one and bi-continuous transformation of $E$ into itself which transforms $a$ into $b$)

Problem of MM. B. Knaster and C. Kuratowski.

Is a set ordered (linearly) without jumps or gaps and such that any set of its intervals (containing more than one element) does not overlap with each other is at most innumerable, necessarily a (ordinary) linear continuum?

Problem of M. M. Souslin.

Is there a decomposition of an interval into $\aleph_{1}$ (non-empty) sets without common points that are $B$ measurable?

Problem of M. W. Sierpinski.

Does there exist an uncountable linear set $E$ such that every linear set homeomorphic to $E$ has zero Lebesgue measure? Can we prove the existence of such a set assuming that $2^{\aleph_{0}} = \aleph_{1}$?

Problem of M. W. Sierpinski.

Can we find without the hypothesis of the continuum ($2^{\aleph_{0}} = \aleph_{1}$) that a sum of $\aleph_{1}$ sets of zero Lebesgue measure is not necessarily zero Lebesgue measure? that a sum of $\aleph_{1}$ sets of the first category is not necessarily of the first category? that a product of $\aleph_{1}$ sets ($A$) is not necessarily a set ($A$)?

Problem of M. W. Sierpinski.

Can we establish without the hypothesis of the continuum the existence of a planar set which has zero (Lebesgue) measure on any direction parallel to the abscissa axis and whose complement has zero measure on any direction parallel to the ordinate axis?

Problem of M. H. Steinhaus.

Can we give an effective example of a set of real numbers $E$ such that any sum, difference, product or quotient of two numbers of $E$ (except division by 0) belongs to $E$ and that $E$ is uncountable, but distinct from the set of all real numbers?

Problem of M. S. Mazurkiewicz.

What is the power of sets complementary to sets ($A$)?

Problem of M. N. Lusin.

Notice. The linear sets ($A$) are orthogonal projections (on a line) of the measurable planar sets $B$. M. Lusin demonstrated that the power of an uncountable set complementary to a set ($A$) is $\aleph_{1}$ or $2^{\aleph_{0}}$, but we do not know if it can really be $\aleph_{1}$ (in the case where $2^{\aleph_{0}} > \aleph_{1}$).

Does there exist a second class function that is not a limit of almost everywhere continuous functions? Can we give an effective example of a function which is not a limit of piecewise continuous functions?

Problem of MM. T. Feosztyn and W. Sierpinski.

Does there exist a class ($\mathcal{L}$) of M. Fr'echet (i.e. a class in which the limit is defined) of power greater than the continuum, such that any uncountable set of elements of this class contains at least one condensation element?

Problem of M. W. Sierpinski.

Does a (linearly) ordered set of which all well-ordered subsets (increasing and decreasing) are at most countable, necessarily have a power not greater than the continuum?

Problem of M. W. Sierpinski.

Does there exist a planar closed set which is not the sum of two closed sets without common points, but a sum of countable closed sets without common points.

Problem of M. W. Sierpinski.

Does a continuum in $m$-dimensional space which is homeomorphic to any continuum contain necessarily a simple arc (i.e. image of an one-to-one and continuous mapping of the interval $(0,1)$?

Problem of M. Mazurkiewicz.

Does there exist a continuum of which every everything under continuous mapping is indecomposable? (A continuum is said to be indecomposable when it is not a sum of two continua different from it.)

Problem of MM. Knaster and Kuratowski.

Does there exist a continuum (unbounded) which is a sum of its proper saturated sub-continuums where any two of them do not intersect?

(We say that a real subcontinuum $K$ of $C$ is saturated, when there exists no continuum different from $K$ and $C$ which contains $K$ and which is contained in $C$.)

Problem of M. Kuratowski.

What is the power of the set of all values that a Baire class 1 function does not take?

(This problem is equivalent to problem 9 of M. Lusin, t. I. p.224. It would be enough to solve this problem for functions admitting uncountable points of discontinuity.)

Is a (linear) set of power less than the continuum necessarily of the first category of M. Baire?

Problem of M. Ruziewicz.

Does there exist in each biconnected set $B$ a point $p$ such that the set $B - (p)$ contains no connected set?

Notice. According to a theorem of M. Kline (this volume, p.238), there cannot exist in a connected set $B$ more than one point $p$ enjoying the property in question. We know, on the other hand, that, if such a point exists, the set $B$ is biconnected, i.e. it is not the sum of two disjoint connected sets containing more than one point (cf. Knaster and Kuratowski, Fund. Math. II, p, 214.)

Problem of M. Kuratowski.

Let $f(E)$ be a function defined by any measurable ($L$) set $E$ of a Euclidean space of $m \geq 3$ dimensions and satisfying the following conditions:

$f(E) \geq 0$.
$f(E_{0}) = 1$ for a certain set $E_{0}$ of measure 1.
$f(E_{1} +E_{2}) = f(E_{1}) + f(E_{2})$, if $E_{1} E_{2} = 0$.
$f(E_{1}) = f(E_{2})$, if $E_{1}$ and $E_{2}$ are superposable.

Does the function $f(E)$ necessarily coincide with the Lebesgue measure of the set $E$?

(For $m=1$ and $m=2$ the answer is negative, as M. Banach proved in a memoir which will be published in the volume IV of this journal.)

Problem of M. Ruziewicz.

Given a set of real numbers which is not of the first category in any interval, is there a decomposition: $A = B + C$, $B \times C = 0$ such that neither $B$ nor $C$ are of the first category in any interval?

Notice. We could give the affirmative answer in the hypothesis additional that $A$ has the Baire property (in the sense established in this volume), p.319 . M. Sierpinski also pointed out the affirmative answer in the hypothesis of the continuum, $\aleph_{1} = 2^{\aleph_{0}}$.

Problem of M. Kuratowski.

Let us call the (linear) set $E$ perfectly measurable, if every set homeomorphic to $E$ is measurable in the sense of Lebesgue. What is the power of the class of perfectly measurable sets? Is a complementary set to a perf. measurable set always perf. measurable?

Problem of M. Urysohn.

Does there exist a function of a real variable $ f(x)$ pantachically dicontinuous and such that we have for all real $x$

\[\lim_{h \to 0} \frac{f( x+h ) - f(x-h)}{2h} = 0?\]

Problem of M. Steinhaus.

Is a function satisfying Baire's condition necessarily measurable ($L$)? What is the power of all the functions of a real variable satisfying the Baire condition? (We say that a function $f(x)$ satisfies the Baire condition, if it is continuous on any perfect set when we neglect the sets of the first category has with respect to that perfect set.)

Problem of M. Sierpinski.

Can a planar set, such that any straight line meets it at two (and only two) points, be measurable ($B$)? (The existence of such a set has been demonstrated, using M. Zermelo's theorem, by MM. Mazurkiewicz¹ (in 1914) and Rosenthal² (in 1922))

Is a class 3 function of M. Baire always a superposition of three class 1 functions, that is to say, does it exist for any function $f(x)$ of class 3 three class 1 functions $\varphi(x)$, $\psi(x)$ and $\vartheta(x)$, such that we have for all real $x$

\[f(x) = \varphi \{\psi[\vartheta (x)]\}\]

Problem of M. Lusin.

Is the set $D(E)$ of the distances of the points of a linear set $E$ that is ($B$) measurable always ($B$) measurable? ($D(E)$ is therefore the set of all numbers $|x - y|$, where $x$ and $y$ belong to $E$. We can prove the existence of set $E$ that are measurable ($L$), such that $D(E)$ is non-measurable ($L$).)

Problem of M. Sierpinski.

If $E$ is a ($B$ measurable) planar set, let us denote by $N(E)$ the set of all real numbers $a$, such that the line $x = a$ meets $E$ at a uncountable infinite number of points. Is the set $N(E)$ necessarily a set ($A$), or, more simply, is it measurable ($L$)? (It can be shown that the set of all real numbers $a$, such that the line $x = a$ meets the ($B$ measurable) set $E$ in an infinity number of points is always a set ($A$).)

Problem of M. Sierpinski.

Let $F$ be a planar set, p. ex. closed (or, more generally, measurable) - A point $x$ of $F$ will be said to be linearly accessible if there exists a rectilinear segment $\overline{xp}$ such that all its points (except the point $x$) are outside $F$. Can it be shown that the set $A$ of all linearly accessible points of $F$ is always measurable ($L$)?

Problem of M. Urysohn.

If $f(x)$ is an arbitrarily given function (measurable or not), what is the measure of the sum of all points $x$, such that

\[\lim \limits_{h\to 0} \left |\frac{f( x+h ) - f(x)}{h}\right| = \infty \]

Problem of M. Ruziewicz.

Is the statement "$\mathsf{m} = 2\cdot\mathsf{m}$ for any transfinite cardinal number $\mathsf {m}$" equivalent to the axiom of choice?

Cf. my Note "On some theorems which are equivalent to the axiom of choice" in this volume, p. 147.

Problem of M. Tajtelbaum-Tarski .

Is a planar closed set, where every point is linearly accessible, necessarily of zero surface measure?

Problem of M. Banach.

A point $x$ of $F$ is said to be linearly accessible if there exists a rectilinear segment $\overline{xp}$ such that all its points (except the point $x$) are outside $F$. M. Urysohn proved that the set of all linearly accessible points of a planar closed set is always a set ($A$) of M. Souslin, but may not be measurable ($B$). But we do not know if the set of all linearly accessible points of a planar $G_{\delta}$ set is measurable ($L$) (Cf. Problem 29, Fund. Math. t. V, p. 337).

Is the image of a one-to-one and continuous map (in one sense) of a set complementary to a set ($A$) of M. Souslin necessarily a homeomorphism?

Problem of Sierpinski.

Let us call a (linear) set ($B$) measurable of class $\alpha$ irreducible if it is not of class $<\alpha$ in any interval. What is the power of the set of all topological types of irreducible sets of class $\alpha$?

Let us call a set ($A$) irreducible, if it is not ($B$) measurable in any interval. What is the power of all the topological types of the sets ($A$) irreducible?

Problem of MM. Alexandroff and Urysohn.

Let us call the (linear) set $E$ perfectly measurable in the narrow sense, if every unambiguous and continuous image of $E$ is measurable in the sense of Lebesgue. Is a set complementary to a set perfectly measurable in the narrow sense always a homeomorphic image?

Cf. Problem 22 of P. Urysohn (Fund. Math. t. IV, p. 368), solved by M. Lavrentieff (Fund. Math. t. VI, p. 159).

Probleme de M. O. Nikodym.

According to M. Souslin, if $E$ is a set ($A$) and $H$ a complementary set to a set ($A$), and if $E \subset H$, there exists a set $Q$, ($B$) measurable, such that $E\subset Q \subset H$³. Does this proposition admit a reciprocal, that is to say, if $E$ is a complementary set to a set ($A$) and $H$ ------ does a set ($A$), such that $E \subset H$, exist always a ($B$) measurable set $Q$, such that $E \subset Q \subset H$?

Problem of M. Sierpinski.

Is a Jordan (bounded) continuum which contains only a simple closed curve homeomorphic to one of its (real) sub-continuums?

Problem of M. Zarankiewicz.

Can a square and a circle whose areas are equal be decomposed into a finite number of respectively congruent disjoint subsets?

Problem of M. Tarski.

Does there exist a planar closed set for which the set of linearly accessible points is non-measurable ($B$)? (In the space the problem is resolved in the affirmative sense).

Problem of M. O. Nikodym.

If $E$ is a planar $G_{\delta}$ set, is the set of all real numbers $a$, such that the line $x=a$ meets the set $E$ in one and only one point, not necessarily complementary to a set ($A$) of M. Souslin?

Problem of M. Sierpinski.

If $E_{1}, E_{2}, E_{3}, \dots$ are a sequence of countable linear sets each of which is a projection of a planar set complementary to a set ($A$) of M. Souslin, is the set $E_{1}E_{2} E_{3} $ of the same nature?

Problem of M. Sierpinski.

Does there exist in every continuum $A$ a continuum $B$ such that the set $A - B$ is connected?

Problem of MM. Knaster and Zarankiewicz.

Let $D$ denote a closed set homeomorphic to a planar set located in 3-dimensional Euclidean space, is every point of $D$ is accessible in this space?

(A point $d$ of $D$ is said to be accessible in $E$, when there exists a continuum $C \subset E$ such that $(d) = CD$).

Problem of M. Knaster.

Unfortunately, Problem 44 and 45 by MM. N. Lusin and H. Steinhaus which should have appeared in XI, p.308, were missing.

It is asked to set up the logical relations between the various concepts of homogeneity, in so far as they refer to locally compact sets. In particular, even if the sets are considered to be connected and (or) locally connected.

(See D. van Dantzig , "On topologically homogeneous continua, volume 15, pp. 102, 103).

Problem of M. van Dantzig.

Is every (connected, unbounded) $n$-dimensional manifold involutory homogeneous?

(Cf. D. van Dantzig, le p. 104, ^7.)

Problem of M. van Dantzig.

Let us call a topological group monothetic if an infinite cyclic group is dense in it (in which case it is commutative and can be written additively), and complete if every sequence $x_{} $, which satisfies the Cauchy's convergence criterion $\lim (x_{\nu} - x_{\mu }) = 0$ has a limit element in the group, the question is whether a monothetic group can be complete without being compact.

(See lc p. 116 {}^{29}a))

Problem of M. van Dantzig.

Let $X$ and $Y$ be two Peano continua (= continuous images of the interval) and $Z$ be their topological product (= the space of all pairs $z = (x,y) $ or $\lim z_{n}=z$ when $\lim x_{n} = x$ and $\lim y_{n} = y$).

1^0. If the continuum $X$, as well as $Y$, has the property that in each continuous transformation of this continuum into a subset there exists an invariant point, is it true that $Z$ has the same property?

2^0. If the continuum $X$, as well as $Y$, is uni-consistent (= in each decomposition of this continuum into two sub-continuums the common part of these sub-continuums is connected), is it true that $Z$ is uni-coherent?

Problem of M. Kuratowski.

Is the topological circle the only homogeneous locally connected curve? (Curve = one-dimensional connected compact space. A space is called homogeneous, if for each of its two points $p$ and $q$ there exist a homeomorphism which sends $p$ to $q$ ). On the plane, the circle is the only homogeneous locally connected curve.

(Cf. Mazurkiewicz, Fund. Math. V, p. 137).

Problem with M. K. Menger.

Are there any or even an infinite number of compact one-dimensional spaces, any two of which are incomparable in dimension one? Two spaces $R$ and $R^{\prime}$ may be called one-dimensionally incomparable, if no one-dimensional subset of $R$, (or of $R^{\prime}$) is homeomorphic with a subset of $R^{\prime}$ (or of $R^{\prime}$). For example, a line and a continuum without partial arcs are incomparable in one dimension. (If there are $n$, or $\aleph_{0}$ in pairs of one-dimensionally incomparable curves, then there exist compact one-dimensional spaces which contains at least $2^{n} +1$, or $2^{\aleph_{0}}$ monotone, $F_{\sigma}$-additive, topological, compactifiable systems of subsets).

(See Monthly Issues f. Math. and Phys. 36, p. 207).

Problem of M. K. Menger.

Does there exist a continuum of which every other continuum is a continuous image?

Problem of M. H. Hahn.

1. Can any absolute retract be decomposed into a finite number of absolute retracts with arbitrarily small diameters?

Can any $R$ set be decomposed into a finite number of absolute retracts?

(The definition of absolute retracts and $R$ sets is given, for example, in Fund. Math. XIX, p. 222).

Is every partial continuum $C$ of the Euclidean $n$-dimensional space $R_{n}$, which intersects $R_{n}$ and which is transferred by arbitrarily small transformations (i.e. by a continuous mapping which sends every point from $C$ to an arbitrarily close point of $R_{n}$) into a subset of $R_{n}$ that is foreign to it, a ($n-1$) dimensional space?
Can every route image lying in $R_{3}$ and intersecting $R_{3}$ be mapped continuously without fixed points?

Problem of M. K. Borsuk.

Let $A$ and $B$ be two topological spaces and $A^{2}$ and $B^{2}$ respectively their squares (a. `ad $A^{2}$ p. ex. consists of all your pairs ($a_{1}$, $a_{2}$) extracted from $A$).

Is it true that if $A^{2}$ and $B^{2}$ are homeomorphic, are $A$ and $B$ too?

In case of a positive answer, we deduce that, if $C$ is a set which is not homeomorphic to none of $C^{n}$, $n>1$, the sets $C^{m}$ and $C^{n}$ are neither homeomorphic for $m \neq n$; this provides in the case where $C$ is an interval of the theorem of "invariance of the dimension" of M. Brouwer.

Problem of M. S. Ulam.

Does there exist an infinite-dimensional continuum that does not contain any finite-dimensional continuum?

Problems of M. S. Mazurkiewicz.

Does there exist in a set $E$ of size $\aleph_{1}$ a countable system of subsets $A_{1}, A_{2},\dots$ such that one has the form \[X = \overline{\lim} A_{p_{n}}\] where ($p_{1}, p_{2},\dots$ is a subsequence of the natural numbers, $\overline{\lim}$ means the Borel complete limit set) all subsets $X, A_{1}, A_{2},\dots$ are received from $E$?

(It is a question of proving the negation without using the continuum hypothesis).

Problems with M. F. Hausdorff.

A function with the Baire property (i.e. continuous on any perfect set, when we neglect a set of $1^{st}$ category relative to this set) of a function with the Baire property, is it of the same nature?

Problem of M. W. Sierpinski.

Let $\rho$ be the set of all homeomorphic transformations of the Cartesian plane in itself, of the form: \[x^{\prime} = x, \quad y^{\prime} =f( x,y )\] and \[x^{\prime}=g( x,y ), \quad y^{\prime}=y.\]

Let us denote by $\sigma$ the group formed by all the finite superpositions of all the transformations belonging to $\rho$. Can an arbitrary homeomorphic transformation of the plane in itself always be approached by those of the group $\sigma$?

An analogous problem remains for $n>2$ dimensional spaces.

Problem of M.S. Ulam.

Let $E$ be a $G_{\delta}$ set in the plane (more generally: a Borel set) all of whose intersections with the lines parallel to the $y$ axis are closed sets (more generally: $F_{\sigma}$). Is the projection of $E$ on the $x$ axis always a Borel set?

Problem of M. E. Szpilrajsn.

The (real) function $f(x)$ of the real variable $x$ is called symmetric-continuous if for every $x$ \[\lim \limits_{h \to 0} [f( x+h ) - f(x - h)] = 0.\] Can the set of discontinuities of such a function be uncountable? Can it be an arbitrarily prescribed set $F_{\sigma}$? (That they are an arbitrarily prescribed countable set is easy to see.)

Problem of M. F. Hausdorff.

Two compact spaces $A$ and $B$ have the same homotopy type, when there exists a continuous transformation $f$ from $A$ to $B$ and a continuous transformation $\varphi$ from $B$ to $A$, such as the superpositions $\varphi f$ and $f\varphi$ (consider respectively as transformations of $A$ into $A$ and of $B$ into $B$) are homotopic to the identity. Are two closed varieties of the same homotopy type always homomorphic?

Problem of M. W. Hurewicz.

Do there exist two in $R^{n}$ orientable Manifolds $M_{1}^{k}$ and $M_{2}^{k}$, whose Complementary $R^{n} - M_{1}^{k}$ and $R^{n} - M_{2}^{k}$ are homomorphic and whose Homology rings are not isomorphic?
Let $B_{0}, B_{1}, B_{2}, \dots, B_{\omega}, \dots, B_{\alpha}, \dots$ be Borel classes of sets, formed in starting from any class of abstract sets. We know that $B_{\alpha} = B_{\alpha+}$. results in $B_{\alpha}= B_{\beta}$ for all $\beta>\alpha$; let $\alpha_{0}$ be the first number $\alpha$ satisfying this condition. What are the numbers $\nu$ for which there exist classes $B_{0}$ such that we have $\alpha_{0} = \nu$? (Cf. Fund. Math. t. XV, p. 284).

Problem of M. A. Kolmogoroff.

Does the weak $LC$ property entail the strong $LC$ property for any compact metric space? Same question for $HLC$ properties . (For definitions see Annals of Mathematics , vol. 85, p. 119-129 and Duke Mathematical Journal, vol . 1, p. 1-18).

Problem of M. S. Lefschetz.

Is the property ($C$) of linear sets invariant with respect to homeomorphic transformations and, more generally, with respect to continuous transformations? (We say that a set $E$ has the property ($C$), when there exists for each sequence $\{a_{n}\}$ of positive numbers a decomposition $E=E_{1}+E_{2} + $ such that the diameter of $E_{n}$ does not exceed $ a_{n}$ for $n= 1, 2,\dots$. Cf. Fund. Math, volume XI, p. 304; volume XV, p. 126; volume XXII, p. 310.)

Problem of M. W. Sierpinski.

If $E_{1}$ and $ E_{2}$ are two linear sets always of first category (i.e. of first category on any perfect set), the set $E_{1} \times E_{2}$. (i.e. the set of all points $(x, y)$ of the plane where $x \in E_{1}$ and $y \in E_{2}$) Is it of the same nature?

Problem of M. E. Szpilrajn.

When a Jordan curve (in a three-dimensional space) has a determinate tangent at each point, does there necessarily exist a parametric representation of this curve expressing the cartesian coordinates of a point of this curve as functions which can be derived from a parameter (and in this case the three derivatives cannot all cancel out at the same time)?

If the answer is negative, the question is asked again by admitting a set of zero measurement values of the parameter where the imposed conditions are not both satisfied.

Problem of M. M. Fr'echet.

Does there exist a linear set $ E$ such that each linear analytical set is a one-to-one and continuous (in one sense) image of $E$?

Problem of M. W. Sierpinski.

Does there exist an infinite sequence $S$ of functions of a real variable (measurable or not), such that any function of a real variable of class $2$ Baire is a limit of some sequence extracted from $S$?

(According to M. C. Burstin such a sequence $S$ cannot be composed uniquely of measurable functions⁴)

Problem M. W. Sierpinski.

Does there exist in the $n$-dimensional Cartesian space ($n>1$) a set always of the first category (i.e. of first category on each perfect set) and which is of positive dimension?

(M. W. Hurewicz demonstrated using the continuum hypothesis that there exists in Hilbert space an uncountable set $H$ of which each uncountable subset is of infinite dimension⁵). M. F. Hausdorff noticed that the set $H$ is always of the first category. This follows easily from the fact that each separable metric space $M$ is the sum of a set of dimension $0$ and a set of first category in $M$. -- There therefore exists, if $2^{\aleph} = \aleph_{1}$, in Hilbert space a set always of first category and of positive dimension).

Probleme de M. E. Szpilrajn.

Let, in three-dimensional Euclidean space, $E$ be a homeomorphic image of the solid sphere $S$ and $L$ be a rectilinear segment whose interior is contained in the interior, and the ends in the border of $E$ . Does there still exist a homeomorphism transforming $E$ into $S$ such that $L$ is transformed into the diameter of $S$?

Problem of M. K. Borsuk.

Let $\mathbf {B}(\mathbf {F})$ be the smallest family of sets containing the given family $ $ and closed with respect to the operations $\sigma$ and $\delta$ (addition and multiplication of countable values). Is there a family of enumerable $\mathbf{D}$ of sets, such that all linear analytic sets belong to $\mathbf {B}(\mathbf{D})$ ?

Problem of M. S. Ulam.

Is there an infinite set $E$ (e.g. the set of all natural numbers) and a function $f(X)$ which maps to any subset $X$ of $E $ a subset $f(X)$ of $E$, so that:

$X \subset f(X)$ for $X \subset E$,
$f(X+Y) = f(X) + f(Y)$ for $X \subset E$, $Y \subset E$,
there exists for every set $Y \subset E$ at least one set $X \subset E$, such that $Y = f(X)$,
there exists at least one set $X_{0} \subset E$, such that $f(X_{0}) \neq X_{0}$.

If we replace the relative condition to $f(X+Y)$ by the condition weaker than $f(X) \subset f(Y)$ pure $X \subset Y \subset E$, the positive answer is obvious.

Problem of M. E. $\check{\text{C}}$ech.

Problems resolved.

When is a set of points $P$ a one-to-one and continuous image (in a sense) of $Q$ and $Q$ a one-to-one and continuous image of $P$, can we affirm that the sets $P$ and $Q$ are homeomorphic?

Negative solution from M. Kuratowski, Fund. Math. t. II, pp. 158-160

Is there a decomposition of an interval into $\aleph_{1}$ measurable sets ($B$), each non-empty and without common points?

Affirmative solution from MM. Lusin and Sierpinski, Comptes Rendus, t. 175, p.357 (note of August 21, 1922).

Third part. Can it be shown that a product of $\aleph_{1}$ sets ($A$) is not necessarily a set ($A$)?

Affirmative solution from MM. Lusin and Sierpinski, Journ, de Math. 1923 (The authors define a set which is complementary to a set ($A$) is not necessarily a set ($A$).)

Can we give an effective example of a set of real numbers $E$, such as any sum, difference, product or quotient of two numbers of $E$ (except division by 0) belongs to $E_{1}$ and that $E$ is uncountable, distinct from the set of all real numbers?

Affirmative solution from M. Souslin, Fund. Math. IV, p.311.

First part. Is there a second class function which is not the limit of almost everywhere discontinuous functions?

Affirmative solution from M. Zalcwasser.

Does an ordered set of which all well-ordered subsets (increasing or decreasing) are at most enumerable necessarily have a power no greater than that of the continuum?

Affirmative solution from M. Urysohn, Fund. Math. V (to appear).

Is there a continuum of which any subcontinuum is indecomposable?

Affirmative solution from M. Knaster, Fund. Math. III pp. 247-286.

Does there exist a continuum which is a sum of its proper disjoint saturated subcontinua?

Affirmative solution by MM. Knaster and Kuratowski, Fund. Math. V.

Accounts of the Soc. of Varsovic Sciences , t. VII, p.382.↩︎
Sitzungaber d. Bayer. Akad. d. Wiss., math.-phys. K1, 1922, p. 223.↩︎
See p. e. N. Lusin and W. Sierpinski Journ. of Math. t. II (1923) p. 60; also Bull. Acad. Krakow 1918 p. 40.↩︎
Monatshefte f. Math. u. Phys. 28 (1917), p. 107.↩︎
Fund . Math. 19 (1932), p.8.↩︎