Sequence covering map

In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties (often, the spaces being Hausdorff and first-countable is more than enough) then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness (whenever such characterizations hold).

Definitions

Preliminaries

Main articles: Sequential space and Fréchet–Urysohn space

A subset $S$ of $(X,\tau )$ is said to be sequentially open in $(X,\tau )$ if whenever a sequence in $X$ converges (in $(X,\tau )$) to some point that belongs to $S,$ then that sequence is necessarily eventually in $S$ (i.e. at most finitely many points in the sequence do not belong to $S$). The set $\operatorname {SeqOpen} (X,\tau )$ of all sequentially open subsets of $(X,\tau )$ forms a topology on $X$ that is finer than $X$'s given topology $\tau .$ By definition, $(X,\tau )$ is called a sequential space if $\tau =\operatorname {SeqOpen} (X,\tau ).$ Given a sequence $x_{\bullet }$ in $X$ and a point $x\in X,$ $x_{\bullet }\to x$ in $(X,\tau )$ if and only if $x_{\bullet }\to x$ in $(X,\operatorname {SeqOpen} (X,\tau )).$ Moreover, $\operatorname {SeqOpen} (X,\tau )$ is the finest topology on $X$ for which this characterization of sequence convergence in $(X,\tau )$ holds.

A map $f:(X,\tau )\to (Y,\sigma )$ is called sequentially continuous if $f:(X,\operatorname {SeqOpen} (X,\tau ))\to (Y,\operatorname {SeqOpen} (Y,\sigma ))$ is continuous, which happens if and only if for every sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X$ and every $x\in X,$ if $x_{\bullet }\to x$ in $(X,\tau )$ then necessarily $f\left(x_{\bullet }\right)\to f(x)$ in $(Y,\sigma ).$ Every continuous map is sequentially continuous although in general, the converse may fail to hold. In fact, a space $(X,\tau )$ is a sequential space if and only if it has the following universal property for sequential spaces:

for every topological space $(Y,\sigma )$ and every map $f:X\to Y,$ the map $f:(X,\tau )\to (Y,\sigma )$ is continuous if and only if it is sequentially continuous.

The sequential closure in $(X,\tau )$ of a subset $S\subseteq X$ is the set $\operatorname {scl} _{(X,\tau )}S$ consisting of all $x\in X$ for which there exists a sequence in $S$ that converges to $x$ in $(X,\tau ).$ A subset $$