This is the list of open problems in topological algebra posed on the
conference dedicated to the 20th anniversary of the Chair of Algebra and
Topology of Lviv National University, that was held on 28 September 2001.

A topological group $G$ is called an $M_\omega$-group if it admits a
countable cover $\K$ by closed metrizable subspaces of $G$ such that a subset
$U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every
$K\in\K$. It is shown that any two non-metrizable uncountable separable
zero-dimenisional $M_\omega$-groups are homeomorphic.

For a topological monoid S the dual inverse monoid is the topological monoid
of all identity preserving homomorphisms from S to the circle with attached
zero. A topological monoid S is defined to be reflexive if the canonical
homomorphism from S to its second dual inverse monoid is a topological
isomorphism. We prove that a (compact or discrete) topological inverse monoid S
is reflexive (if and) only if S is abelian and the idempotent semilattice of S
is zero-dimensional. For a discrete (resp. compact) topological monoid its dual
inverse monoid is compact (resp. discrete).

Let X be a locally compact Polish space and G a non-discrete Polish ANR
group. By C(X,G), we denote the topological group of all continuous maps f:X
\to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup
consisting of all maps with compact support. It is known that if X is compact
and non-discrete then the space C(X,G) is an l_2-manifold.

We survey some properties of homotopical and homological $Z_n$-sets in
topological spaces.

A metric space $M$ us said to have the fibered approximation property in
dimension $n$ (br., $M\in \mathrm{FAP}(n)$) if for any $\epsilon>0$, $m\geq 0$
and any map $g: I^m\times I^n\to M$ there exists a map $g':I^m\times I^n\to M$
such that $g'$ is $\epsilon$-homotopic to $g$ and $\dim g'\big(\{z\}\times
I^n\big)\leq n$ for all $z\in I^m$. The class of spaces having the
$\mathrm{FAP}(n)$-property is investigated in this paper. The main theorems are
applied to obtain generalizations of some results due to Uspenskij and
Tuncali-Valov.

We study algebraic and topological properties of topological semigroups
containing a copy of the bicyclic semigroup C(p,q). We prove that each
topological semigroup S with pseudocompact square contains no dense copy of
C(p,q). On the other hand, we construct a (consistent) example of a
pseudocompact (countably compact) Tychonov semigroup containing a copy of
C(p,q).

Suppose G is a topological group containing a (closed) topological copy of
the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal
k-space) then every closed metrizable subset in $G$ is locally compact.
Applying this result to topological groups whose underlying topological space
can be written as a direct limit of a sequence of closed metrizable subsets, we
get that every such a group either is metrizable or is homeomorphic to the
product of a $k_\omega$-space and a discrete space.

An invariant ideal I on a group G is defined to be Pack_n-complete if it
contains each subset A of G with infinite packing index Pack_n(A). We prove
that the ideal of absolute null subsets of an amenable group and the ideal of
small subsets of an abelian group are Pack_n-complete for every n>1. Also we
show that each invariant ideal I on an amenable group has the packing
completion Pack_n(I) (which is the smallest Pack_n-complete ideal containing
I).

We detect Hilbert manifolds among homogeneous metric spaces and apply the
obtained results to recognizing Hilbert manifolds among homogeneous spaces of
the form G/H where G is a metrizable topological group and H is a closed
balanced subgroup of G.