A. V. Arhangel¢skii

Some problems and lines of investigation in General Topology

Chair of General Topology and Geometry, Department of Mathematics, Moscow State University, USSR

26.8.1988

Introduction

Some of the problems under consideration belong to relatively new (or completely new) directions of investigation. In such cases we supply the reader with necessary definitions and relevant basic facts.

There is no single rule explaining how the problems appearing in the article were chosen. One easily imagines another article of the same kind with the disjoint set of problems of no lesser importance.

Nevertheless, I consider the problems chosen to be quite interesting and difficult to solve. Many of the problems involve Lindelöf spaces. One gets an impression that we know too little about this nice class of spaces.

In Sections 1.4 and 2.8 some new results on Lindelöf spaces are established.

Notations and terminology are as in [8] and [21]. Compact means bicompact. \mathbbN⁺ is the set of all positive integers, all spaces are assumed to be T₁. Regularity is included in the definition of Lindelöf space.

1  Problem on continuous mappings

1.1  Mappings and classification of spaces

A very general classical problem embracing many concrete interesting problems is: when every space from a given class P can be represented as an image of a space from a class R (more narrow than P) under a mapping satisfying certain preformulated conditions. Here is an old open problem of this kind (see [8]).

Problem 1

A motivation for Problem 1 can be seen in the following simple fact: if f:X ® Y is an open continuous mapping with compact preimages of points such that f(X)=Y and X is paracompact then Y is metacompact.

Problem 1 was for the first time formulated by myself in 1965 at Moscow Seminars on Topology. It appeared in print in [8] and was mentioned in the Nice Congress lecture in 1970.

It is known (H. H. Wicke, see [8]) that there exists a Hausdorff metacompact space Y which cannot be represented as an image of a paracompact Hausdorff space under an open continuous mapping with compact preimages of points. On the other hand every space with a uniform base is an image of a metrizable space under such a mapping (S. Hanai, A. Arhangel¢skii, see [8], [21]).

The following version of Problem 1 should be also mentioned:

Problem 2

H. Junnilla has established a non-trivial fact: if f:X ® Y is such a mapping as in Problem 2, and f(X)=Y, X is a paracompact Hausdorff space and Y is Hausdorff then Y is metacompact. Thus the question is whether this theorem can bp reversed.

The following question asked by A.V. Arhangel¢skii at Moscow University six or seven years ago appears in print for the first time.

Problem 3

The last condition means that all points are G_d's. It is easily seen that if X is a Lindelöf space of countable pseudocharacter, Y is a Tychonoff space and f:X ® Y is a one-to-one continuous mapping then Y is also a space of countable pseudocharacter. On the other hand the class of hereditarily Lindelöf spaces is preserved by continuous mappings (onto Tychonoff spaces). There is no known restriction on the cardinality of Lindelöf spaces of countable pseudocharacter except that it has to be a non-measurable (in Ulam's sense) cardinal (see [8]) - hence the restriction on cardinality in Problem 3.

I do not know what is the answer to Problem 3 in the case of compact Hausdorff images. For example we can consider the following

Problem 4

To get the positive answer to Problems 3 and 4 would be extremely difficult as we do not know whether in ZFC can exist a Lindelöf space of countable pseudocharacter and of cardinality greater than 2^À₀ (see in this respect [1] and [8]). To remove this cardinality obstacle we formulate

Problem 5

The following open problem is also published for the first time.

We start with a few motivating observations. Probably the greatest defect of Lindelöf property lies in the fact that it is not productive. On the other hand compactness being productive also enjoys the incompressibility property - in the sense that one cannot map a compact space onto a different Hausdorff space by a one-to-one continuous mapping. The idea behind the following problem is that the two properties - productivity and incompressibility - might be related to each other.

Problem 6

Recall that a space X is finally compact if every open covering of X contains a countable subcovering. A student of Moscow University A. Jakivcik has constructed a Hausdorff finally compact space X such that for everv one-to-one continuous mapping f:X ® Y onto a Hausdorff space Y the space Y ×Y is not finally compact. Thus in the class of Hausdorff spaces the question similar to Problem 6 is answered negatively. There are many interesting versions of Problem 6. Here are some of them.

Problem 7

Problems 6 and 7 can be formulated for arbitrary number of (possibly different) factors. For example we have

Problem 8

There is another interesting unsolved problem on one-to-one continuous mappings, which involves Lindelöf spaces. One of topological properties which are opposite in many respects to Lindelöfness is pseudocompactness. This can be considered as a motivation for the following.

Problem 9

Let us call entightments one-to-one continuous mappings onto. Observe that only very few spaces can be enlightened onto compact Hausdorff spaces (see [8]).

Problem 9 was formulated by me six or seven years ago at Moscow Seminars in General Topology. Strangely enough it remains unsolved. Now we shall discuss Lindelöf spaces from another standpoint. It is well known that every Lindelöf space is realcompact (in the sense of E. Hewitt and L. Nachbin - see [21]). Every regular space which is a continuous image of a Lindel\"f space is itself Lindelöf and hence realcompact. The problem arises whether the converse is true. Thus we have

Problem 10

This problem was formulated by A. V. Arhangel¢skii and O. G. Okunev in [7]. It is shown in [7] that a regular space X need not be Lindelöf if every Tycnonoff image of X under one-to-one continuous mapping is a realcompact space.

1.2  Projective properties an projective Cech completeness

Let P be a topological property. Following [5] we say that a space X is projectively P if for every open continuous mapping f:X ® Y where f(X) = Y is a separable metrizable space, the space Y must have the property P. In particular a space X is projectively Cech-complete if every separable metrizable image of X under an open continuous mapping is a Cech-complete space. It can be easily deduced from the known result that every Cech-cmplete space is projectively Cech-complete (note that Cech-completeness is not preserved in general by open continuous mappings onto Tychonoff spaces - see [8]).

For a Tychonoff space X we denote by C_p(X) the space of real-valued continuous functions on X in the topology of pointwise convergence (see [15], [16]). If Y is a closed subspace of X then the restriction mapping r:C_p(X) ® C_p(Y), defined by r(f) = f|Y for every f Î C_p(X), is an open continuous mapping of C_p(X) onto the subspace C_p(Y|X) = r(C_p(X)) of the space C_p(Y) (see [15]). If Y is also countable then C_p(Y|X) has a countable base. Thus if C_p(X) has a property P projectively then C_p(Y|X) has the property P provided Y is a countable closed subspace of X. In particular, if C_p(X) is projectively Cech-complete then C_p(Y|X) is Cech-complete which implies that Y is discrete. Thus if C_p(X) is projectively Cech-complete then every countable closed subspace of X is discrete (see [5]). Arguments of this kind provide us with a motivation for the study of projective properties.

The space C_p(X) is seldom Cech-complete - only if X is countable and discrete [5]. And when C_p(X) is projectively Cech-complete?

The answer is unknown.

Problem 11

Here is an interesting concrete question related to Problem 11.

Problem 12

Observe that in the Cech-Stone compactification b\mathbbN of the discrete space \mathbbN of integers all closed countable subspaces are finite and hence discrete. It is evident that all projective properties are preserved by open continuous mappings. A closed subspace of a projectively Cech-complete space need not be projectively Cech-complete. For example, every pseudocwpact space is projectively Cech-complete, and it is well known that each Tychonoff space can be realized as a closed subspace of a pseudocompact space (see [21]). But very little is known on the behaviour of projective properties under products.

Problem 13

The answer to Problem 13 is unknown even in the case when Y is a metrizable compact space (or Y is the unit segment). There seems to be a good chance that the following special version of Problem 13 will be answered in the positive way.

Problem 14

Whilst the last question gives an impression of being rather special, it is closely related to the basic Problem 11. Alongwith projective Cech-completeness it is quite tepting to investigate projective compactness, projective s-compactness and projective finiteness. Some results of that type are mentioned in [6] and [5].

1.3  Cleavable spaces

Let P be a class of topological spaces. We say that a space X is cleavable with respect to P (or P-cleavable) if for every subset A Ì X there exists a space Y Î P and a continuous mapping f:X ® Y such that f(X) = Y and A = f^-1f(A). If a space X is cleavable with respect to the class of all separable metrizable spaces then X is simply called cleavable. These concepts were introduced in [9] and [6]. Again a motivation for the concept of cleavability can be found in the C_p theory: a Tychonoff space X is cleavable if and only if for every realvalued function f on X one can find a countable family A of continuous real-valued functions on X such that f belongs to the closure of A (with respect to the topology of pointwise convergence on \mathbbR^X).

It is natural to consider M-cleavable spaces - the spaces which are cleavable with respect to the class M of all metrizable spaces. Also k-cleavable spaces deserve attention, these spaces are cleavable with respect to the class of all compact Hausdorff spaces.

It is easily seen that all M-cleavable spaces and all k-cleavable spaces are Hausdorff. In M-cleavable spaces all points are G_d's (see [6], [9]). In [9] it is shown that every M-cleavable paracompact p-space is metrizable. One of the most interesting unsolved problems concerning cleavability is the following one:

Problem 15

In an important case the last problem was solved (see [9]): every Lindelöf cleavable space is a space with G_d-diagonal (see 1.4). One can find many other results on cleavable spaces in [9], and [6].

It is not clear whether the following natural problem will have a nice solution:

Problem 16

One should keep in mind that if a space X can be mapped by a one-to-one continuous mapping onto a separable metrizable space (onto a metrizable space, onto a compact Hausdorff space) then it is cleavable (M-cleavable, k-cleavable, accordingly).

For a given space X it might be an interesting problem to find out whether X is cleavable with respect to some class of spaces with much better properties than X. A general version of Problem 16 may be stated in the following way: given a class P of topological (Tychonoff) spaces, characterize the class P^# of all (Tychonoff) spaces which are cleavable, with respect to P. In this direction, one can find some problems and results in [9], [6].

1.4  An approach to cleavability and a theorem on Lindelöf spaces

Let X be a set and A - a subset of X. A family g of subsets of X will be called a separator for A if for every x Î A and every y òX \A there exists P Î g such that x Î P and y Ï P. In the case when X is a topological space, a separator for A Ì X is said to be closed if it consists of closed sets.

Let t be an infinite cardinal number. We shall say that a topological space X is t-divisible if for every A Ì X there exists a closed separator in X of cardiinality not greater than t. For the sake of brevity we call divisible those spaces which are À₀-divisible.

A space X will be called strictly divisible if for every subset A Ì X there exists a countable separator consisting of closed G_d-subsets of X.

These concepts were considered by the author three or four years ago because of their close relationship to the concept of cleavability (see [6], [9]).

Recall that a family E of subsets of X is said to be separating points of X if for every x and y in X, where x ¹ y, one can find B Î E such that x Î B and y Ï B.

Obviously, in T₁-spaces every network and every pseudobase serves as a separator for all subsets. It is also clear that if f:X ® Y is a mapping onto, A Ì Y and g is a separator for A (in Y) then the family { f^-1(P) : P Î g} is a separator for f^-1(A) in X. It follows that if a space X is cleavable then it is strictly divisible. A slightly more general result:

Proposition 1 If a space X is cleavable with respect to the class of all T₁-spaces with a countable closed network then it is divisible.

Another simple assertion is also quite useful.

Proposition 2 If a space X is t-divisible then every subset A of X can be represented as the union of not more than 2^t closed sets in X.

Proof

It is an easy fact that all points in cleavable spaces are G_d's. Thus the theorem that every cleavable Lindelöf space has G_d-diagonal from [9] is a straightforward corollary to the following result.

Theorem 1 If X is a strictly divisible Lindelöf space then the diagonal in X ×X is G_d.

First we shall prove

Theorem 2 . Let X be a Lindelöf space of countable pseudocharacter which is 2^À₀-divisible. Then |X| £ 2^À₀.

Proof

For every B Î E we fix a closed separator g_B such that |g_B| £ 2^À₀.

Then that family g = È{ g_B : B Î E } is separating points of X and |g| £ 2^À₀ ·2^À₀ = 2^À₀. Besides g consists of closed sets. Hence H = { X \P : P Î g} is a pseudobase in X and |H| = |g| £ 2^À₀. Thus pw(X) £ 2^À₀.

By another well known formula (see ([1], [22]) we have:

|X| £ (pw(X))Y(X) ·l(X) £ 2À0À0 ·À0 = 2À0.

Theorem 2 is proved.

Theorem 3 If X is a divisible Lindelöf space of countable pseudocharacter then pw(x) £ À₀ - i.e. there exists a countable pseudobase in X.

Proof

The following assertion is obvious.

Proposition 3 In every strictly divisible space all points are G_d's.

Proof of Theorem 1. By Proposition 3, X is a space of countable pseudocharacter. By Theorem 2, |X| £ 2^À₀. Following the proof of theorem 3, we can construct a countable pseudobase in X consisting of F_s-sets. It is easy to show that every space with a countable pseudobase consisting of F_s-sets has G_d-diagonal.

Remark. Not every Hausdorff space with a countable base has G_d-diagonal

2  More problems on Lindelöf spaces

2.1  The problem of D.V. Rancin

Let us say that a subspace Y of a space X is finally compact in X if every open covering g of X contains a countable subfamily m such that Y Ì Èm. If X is regular and Y is finally compact in X we say that Y is Lindelöf in X. These definitions belong to D. Rancin, who has stated the following problem.

Problem 17

Clearly if Y Ì Z Ì X where X is regular and Z is Lindelöf then Y is Lindelöf in X. Though Problem 17 is seven or eight years old and was discussed several times at seminars and conferences on Topology, it remains unsolved.

Several persons have observed that if Y = X, X is Hausdorff and Y is finally compact in X then it is not true in general that there exists Z Ì X such that Y Ì Z, and Z is finally compact (in itself).

In one particular case Problem 17 was solved: A. V. Arhangel¢skii and Hamdi M. M. Genedi (A.R.E.) have shown that if Y is pseudocompact and Y is Lindelöf in X then [`Y] is compact.

2.2  The Lindelöf property and tightness

Let X be a space. Recall that the tightness of X is countable (notation: t(X) £ À₀) if for every A Ì X and every x Î [`A] there exists a countable set B Ì A such that x Î [`B]. It is known that t(X) £ À₀ does not imply that t(X ×X) £ À₀ (see [12]). On the other hand V. I. Malychin has shown that if X is a compact Hausdorff space and t(X) £ À₀ then t(X ×X) £ À₀ (see [1]). The following question remains unanswered.

Problem 18

Problem 19

A. G. Leiderman and V. I. Malychin have shown that the existence of such X, as in Problem 18 does not contradict ZFC.

2.3  Networkweight and the Lindelöf property

The following problem was formulated by myself ten or twelve years ago. Under CH the answer to it is ``yes'' (see [24]).

Problem 20

Some further material relevant to Problem can be found in [27].

2.4  Lindelöf property in topological groups

In the class of all topological groups many properties behave much better than in the class of all Tychonoff spaces. For example, the first axiom of ountability beitomes the same as metrizability, pseudocompactness becomes productive, countable pseudocharacter is equivalent to the G_d-diagonal property. One of common vices of a great number of very nicely looking topological properties is their unstability with respect to products: the square of a normal space need not be normal, the square of a paracompact space need not be paracowact, the square of a Lindelöf space may be a very non-Lindelöf space - it need not even be normal. So that we would enjoy very much if in the presence of a group structure a topological property would become ``productive''. This is exactly what happens in the case of pseudocompact topological groups (see [19]). For paracompactness, normality, countable compactness and Lindelöf property the situation is far from being clear. To construct two Lindelöf topological groups G and H such that G×H is not Lindelöf one can start with any two spaces X and Y such that Xⁿ and Yⁿ are Lindelöf for all n Î \mathbbN⁺ while the product space X ×Y is not Lindelöf. Under CH such spaces X and Y were constructed by E. Michael, while T. Przymysi\'nski has produced X and Y with the above mentioned properties just in ZFC (see ([24]). Then the free topological groups F(X) and F(Y) are the groups G and H we are looking for. It turns out to be much more difficult to construct a single Lindelöf topological group G such that the square G ×G is not Lindelöf. V. I. Malychin has done it under CH: he has constructed a Lindelöf topological group G such that the space G ×G is not even normal (see [12]). The following question (put forward for the first time in [3]) remains open:

Problem 21

Similar questions about normality, paracompactness and countable compactness also remain unsolved.

The following question is closely related to Problem 21 and is interesting in itself.

Problem 22

A similar question can be asked about normality, pseudocompactness, countable compactness and many other topological properties.

Another interesting problem connected to Problem 21 in the other obvious way is as follows.

Problem 23

The same question can be asked about normal spaces, paracompact spaces and so on.

Let us mention a concrete version of Problem 23.

Problem 24

I have formulated the last two questions at Moscow Seminars on Topology in 1978 and to Eric van Douwen at the time of Moscow International Conference on Topology in 1979. He communicated the questions to other mathematicians and they appeared in Comfort's article [19], who was doubtful to whom to attribute the questions. Problems 23 and 24 were also mentioned in [3].

Clearly Sorgenfrey line is not homeomorphic to a topological group - otherwise being first countable it would have been metrizable. But it is far from being clear what is the answer to the following question:

Problem 25

Positive answer to the last question would imply positive solution of Problem 21.

Problems similar to the following one arise in the theory of transitive actions of topological groups on topological spaces.

Problem 26

2.5  C_p(X) and Lindelöf property

The space C_p(X) of all real-valued continuous functions on a Tychonoff space X always satisfies the countable chain condition. It follows that C_p(X) is paracompact if and only if C_p(X) is Lindelöf. If X is a Lindelöf S-space (in particular if X is compact and C_p(X) is normal then C_p(X) is Lindelöf (E.A. Resnichenko, see [15]). But we do not know how to characterize in terms of X when C_p(X) is Lindelöf.

Problem 27

It is not clear at all whether Problem 27 should have a ``nice'' sclution. But if P is such a property of X as in Problem 27 and P is not an ``artificial'' topological invariant then it is natural to expect that if X is a space with property P then the free topological sum X ÅX of two copies of X is also a space with property P. If this is the case then the following implication would be true: if C_p(X) is Lindelöf then C_p(X) ×C_p(X) is also Lindelöf.

In this way we arrive to the following problem:

Problem 28

The reasoning preceding Problem 28 reveals that whatever topological property Q of C_p(X) we have which can be characterized by a natural topological property of X, there are very good chances that property Q will prove to be productive: if C_p(X) has property Q then C_p(X) ×C_p(X) also enjoys Q.

There are several topological properties Q for which this conclusion holds (see [15], [16]).

The following interesting problem is obviously related to Problem 28.

Problem 29

There exists an infinite compact Hausdorff space X (actually, X = T(w₁ + 1) such that C_p(X) ×C_p(X) is not homeomorphic to C_p(X) (see [15]).

If instead of C_p(X) in Problem 29 we take an arbitrary locally convex linear topological space L, then the answer is negative (W. Marciszewski [23]) - even when L is separable and metrizable.

An important necessary condition C_p(X) to be Lindelöf is that the tightness of Xⁿ has to be countable for every n Î \mathbbN⁺ (M. Asanov, see [10]).

The next problem at first seems a little bit strange: but actually it is not - it is just one of the ways to ask whether the class of all Tychonoff spaces X such that C_p(X) is Lindelöf, is rich enough.

Problem 30

If instead of ``continuous'' we put ``quotient'' in Problem 30 then the answer will obviously be negative: the space C_p(Y) will also have to be Lindelöf.

An interesting open question is connected to the following theorem (see [15], [16]): The tightness of C_p(X) is countable if and only if the space Xⁿ is Lindelöf for every n Î \mathbbN⁺. From this it follows that if X is just Lindelöf then the tightness of C_p(X) need not be countable: for example if X is Sorgenfrey line then t(C_p(X)) > À₀. But what if we consider only compact parts of C_p(X)?

Problem 31

Problem 31 was for the first time formulated by A. V. Arhangel¢skii and V. V. Uspenskij in [17]. See also [15].

If F is dyadic then the answer is ``yes'' (see [17]). Below, a conditional solution to Problem 31 is given but I believe that the answer to Problem 30 should be ``yes'' just in ZFC.

2.6  A conditional solution of Problem 31 and some further discussion of problems

Theorem 4 The following assertion does not contradict ZFC: Let Y be a Lindelöf space and F - a compact subspace of C_p(Y). Then t(F) £ À₀.

Proof

On the other hand the point x^* | Z is in the closure of the set { x_a | Z : a < À₁ } but not in the closure of the sets { x_b | Z : b £ a} for any a < À₁. - this follows from the definition of M Ì Z. Hence the tightness of H is uncountable and w(H) = À₁, t(H) = À₁. From the reasoning of Z. Balogh and A. Dow (see [18], [20]) it follows that it does not contradict ZFC to assume that H contains a topological copy of the space T(w₁ + 1) = { a: a < w₁ } (with the usual topology).

It remains to apply the following assertion:

Proposition 4 If the space T(w₁ + 1) can be embedded into the space C_p(X), then the space X is not Lindelöf.

Proposition 4 is an obvious reformulation of one of the results of A. V. Arhangel¢skii and V. V. Uspenskij (see [17]).

The proof of Theorem 4 is complete.

We shall formulate now a problem closely related to Problem 31.

Let us say that a space Y is colindelöf if there exists a Lindelöf space X such that Y is homeomorphic to a subspace of the space C_p(X). This concept was introduced in [17]. Obviously every subspace of a colindelöf space is a colindelöf space.

Problem 32

In every compact Hausdorff space of uncountable tightness one can easily find a closed subspace which can be mapped continuously onto the space T(w₁ + 1). Proposition 4 implies that the compact space is not colindelöf. Hence the positive solution of Problem 32 would imply the positive answer to Problem 31.

An interesting question was asked by O.G. Okunev:

Problem 33

If C_p(X) is a Lindelöf S-space then the answer is positive - see [15].

If the answer to Problem 33 is in the affirmative then not every space Y can be represented as a continuous image of a space X such that C_p(X) is Lindelöf (see Problem 30). Indeed, to prove the last assertion it is enough to solve negatively the following problem:

Problem 34

Problem 35

Observe that if Y = C_p(X) - i.e. the space C_p(X) itself is of countable pseudocharacter - then |Y| = |C_p(X)| £ 2^À₀ (see [15]).

2.7  Around Velicko's problem

It was independently proved by Ph. Zenor and N. V. Velicko that (C_p(X))ⁿ is hereditarily Lindelöf for every n Î \mathbbN⁺ if and only if Xⁿ is hereditarily separable for every n Î \mathbbN⁺, and that (C_p(X))ⁿ is hereditarily separable for each n Î \mathbbN⁺ if and only if Xⁿ is hereditarily Lindelöf for all n Î \mathbbN⁺ (see [15], [11]).

N. V. Velicko has proved more: if C_p(X) is hereditarily separable then (C_p(X))ⁿ is hereditarily separable for all n Î \mathbbN⁺ (see [15], [11]). A similar question for hereditarily Lindelöf C_p(X) was formulated by him and left open:

Problem 36

A conditional solution of this problem was given by myself. Let us denote by SA the following assertion: every Tychonoff hereditarily separable space is hereditarily Lindelöf. It was shown by S. Todorcevic that SA is compatible with ZFC (see [27]). Let us recall that s(Y) £ À₀ denotes that every discrete subspace of Y is countable. Obviously if Y is hereditarily Lindelöf or hereditarily separable then s(Y) £ À₀. I have proved assuming SA that if s(C_p(X)) £ À₀ then (C_p(X))ⁿ is both hereditarily Lindelöf and hereditarily separable for every n Î \mathbbN⁺.

It is not clear whether a similar result can be proved for arbitrary cardinal t (I venture to suggest that it cannot be proved). Thus we have the following general version of Velicko s problem:

Problem 37

Observe that M. Asanov has shown that if X is a space with G_d-diagonal and C_p(X) is hereditarily Lindelöf then (C_p(X))ⁿ is hereditarily Lindelöf for all n Î \mathbbN⁺ - i.e. Problem 36 has in this case a positive solution (see [10]).

We should also mention here the following assertion published by R. Pol without an explicit proof in [25]:

(P) If X is a separable Tychonoff space such that C_p(X) ×M is Lindelöf for every separable metrizable space M then for every n Î \mathbbN⁺ each discrete subspace of thp space Xⁿ is countable.

It seems to be unknown at the moment whether this assertion holds in ZFC - the proof R. Pol hints at does not work. One can derive Pol 's assertion from SA - so that one cannot construct counterexample working in ZFC. On the other hand if (P) is true then the answer to Problem 36 is positive.

2.8  Some other problems on C_p(X) and Lindelöf property

It is very easy to construct a non-Lindelöf Tychonoff space X such that C_p(X) is Lindelöf. For example one can take as X the S-product of an uncountable family of unit segments. It is a little more difficult to describe a non-normal (pseudocompact) Tychonoff space X such that C_p(X) is Lindelöf. But the following problem (stated by myself at Moscow University two years ago) remains open:

Problem 38

The next problem was formulated in my talk at the V-th Symposium on General Topology in Prague in 1981 (see [14]).

Problem 39

A. Korovin has checked that if X is the Niemytzky plane or X is the Pixley-Roy space on \mathbbR then C_p(X) is not Lindelöf. There is a major problem in the theory of l-invariant properties which concerns Lindelöf spaces and is still open. Recall that Tychonoff spaces X and Y are said to be l-equivalent (t-equivalent) if C_p(X) and C_p(Y) are linearly homemorphic (if C_p(X) and C_p(Y) are homeomorphic as topological spaces).

Problem 40

This problem was formulated by myself in 1980, it appeared in print in [2],[14]. Observe that O.G. Okunev has shown that normality is not preserved by l-equivalence. For a further discussion of l-invariants see [15] and [16].

I would like to conclude this survey of problems on Lindelöf spaces with the following three questions which came to my mind just recently and the degree of difficulty of which is not quite clear to see.

Problem 41

This problem is somewhat related to Problem 38.

Problem 42

This question is connected to results in Section 1.4.

Problem 43

Observe that if X is a pseudocompact space with a uniform base then u(X) is a compact and by a well known result of B. Scott and S. Watson the space X is metrizable (see [28]).

References

[1]

A. V. Arhangel¢ski. Stroenie i klassifikatsia topologicheskikh prostrastv i kardinal nye invarianty. (The structure and classification of topological spaces and cardinal invariants.) Uspekhi Mat. Nauk, 33(6(204)), 29-84, 272, 1978.

[2]

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Footnotes:

¹This article was published as Comment. Math. Univ. Carolin. 29,4 (1988) 611-629. Reprinted with permission.

²In this version, the page numbers have changed from the original and the bibliographic references have been completed.