© 1996, Topology Atlas
A collection $\gamma$ of subsets of a topological space $X$ is called a $k$-network if whenever $K\subseteq U\subseteq X$, $K$ is compact and $U$ is open there is a finite subcollection $\gamma'$ of $gamma$ such that $K\subseteq\bigcup(\gamma')\subseteq U$. The notion of $k$-network has its origin in the paper of E.Michael [M] where he studies spaces with countable $k$-networks. Since then several authors studied spaces with some types of $k$-networks: point-countable, sigma-locally-finite, sigma-hereditarily-closure-preserving etc. Usually these results characterize certain classes of spaces close to metrizable as spaces having some type of $k$--network or closely related to them collections like $cs^\ast$-networks. In [Fo] L. Foged charcterizes closed images of metrizable spaces as Frèchet spaces with sigma-hereditarily-closure-preserving $k$-networks, paper [GMT] contains a characterization of quotient-s-images of metric spaces as spaces having a certain type of point-countable $k$-networks; the latter paper also contains many useful references. These results have been a source of metrization theorems among which is the fact that a compact space with a point-countable $k$-network is metrizable (recall that a collection $\gamma$ is called point-countable if for any $x$ the set $\{\xi \in \gamma | x \in \xi \}$ is countable). This shows, in part, why spaces with point-countable $k$-networks are of special interest (see also [GMT] for other motivations) so here we consider such spaces only. The result mentioned easily leads to a conclusion that studying $k$-spaces with point-countable $k$-networks means actually studying a subclass of sequential spaces. The subclass has many nice properties which prove useful for both constructing examples of sequential spaces and proving theorems about spaces obtained by some natural operations from, say, metrizable ones. As one of the applications of the former type an example of two Frèchet spaces with a sequential product of high sequential order may be mentioned [NS]. The idea of the second type applications will be clear from the next paragraph about sources of spaces with point-countable $k$-networks.
Now let us consider some natural operations which produce spaces with point-countable $k$-networks. The most trivial class of such spaces is the one consisting of spaces whose all compact subsets are finite. $k$-networks are of no help here. On the other hand starting with spaces containing enough compact subspaces (in a sense that those determine the topology; see [GMT] for precise definitions) like metrizable ones and proceeding to apply operations which preserve point-countable $k$-networks quickly takes one to a class of spaces which is very far from the initial one. The following is an incomplete list of such operations; the exact references may be found in [GMT]. Point-countable $k$-networks are preserved by:
(a) quotient maps with separable (compact) fibers and metrizable domain
(b) countable products; this operations easily leads beyond the class of $k$-spaces
(b) perfect maps
(c) subspaces
(d) shrinking a single closed set to a point
Almost every standard topological invariant, from separation axioms to character, fails to be preserved by combinations of these operations though the presense of a $k$-network makes it possible to study the spaces themselves, their images and preimages under some classical types of maps. If we limit ourselves by considering $k$-spaces only the list above may be extended in the following way. $k$-spaces with point-countable $k$-networks are preserved by:
(e) closed maps (see [S1])
(f) quotient maps with countable fibers (but even open maps with compact fibers do not preserve point-countable $k$-networks!)
(g) quotient maps with separable fibers and domains being Frèchet
In [A] it was shown that
(h) a free topological group over a space with a countable $k$-network has a countable $k$-network
Paper [ZP] contains a construction of a sequential non Frèchet topology on any Abelian group. The corresponding topological group may be shown to have a point-countable $k$-network. A metrization theorem for sequential topological groups may be formulated in terms of $k$-networks and sequential order (see [S2]): a sequential topological group is metrizable iff it has a point-countable $k$-network and sequential order less than $\omega1$ (recall that sequential order is an ordinal that measures the number of steps necessary to obtain the closure of a set by adding limits of convergent subsequences). It should be pointed out that in the case of $k$-space some characterizations of point-countable $k$-networks in terms of convergence (see [T]) are usually used rather than the definition mentioned at the beginning. Surveys [G1] and [G2] contain many references concerning applications of $k$-network and similar collections to generalized metric spaces theory.
[Fo] L. Foged, A characterization of closed images of metric spaces, Proc. Amer. Math. Soc., 95, 1985.
[GMT] G. Gruenhage, E.Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math., 113, 1984, 303--332.
[G1] G. Gruenhage, Generalized metric spaces, Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan, 423--501, North-Holland, Amsterdam, 1984.
[G2] G. Gruenhage, Generalized metric spaces, Recent Progress in General Topology, eds. J. van Mill and T.Hu\v sek, Noth-Holland, Amsterdam, 1990.
[M] E. Michael, $\aleph_0$-spaces, J. of Mat. and Mech., 15, 1966, 983--1002.
[NS] T. Nogura and A. Shibakov, Sequential order of product spaces, Topology and its Applications, 65, 1995, 271--285.
[S1] A. Shibakov, Closed mapping theorems on $k$-spaces with point-countable $k$-networks, Comment. Math. Univ. Carolinae, 36, 1995, 77--87.
[S2] A. Shibakov, Metrizability of sequential topological groups with point-countable $k$-networks, preprint.
[T] Y. Tanaka, Point-countable covers and $k$-networks, Topology Proc., 12, 1987, 327--349.
[ZP] Protasov Zelyanuk, Topologies on Abelian groups Math. USSR Izvestia, 37, 1991.