Sequence covering map

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

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

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

The sequential closure in ${\displaystyle (X,\tau )}$ of a subset ${\displaystyle S\subseteq X}$ is the set ${\displaystyle \operatorname {scl} _{(X,\tau )}S}$ consisting of all ${\displaystyle x\in X}$ for which there exists a sequence in ${\displaystyle S}$ that converges to ${\displaystyle x}$ in ${\displaystyle (X,\tau ).}$ A subset ${\displaystyle S\subseteq X}$ is called sequentially closed in ${\displaystyle (X,\tau )}$ if ${\displaystyle S=\operatorname {scl} _{(X,\tau )}S,}$ which happens if and only if whenever a sequence in ${\displaystyle S}$ converges in ${\displaystyle (X,\tau )}$ to some point ${\displaystyle x\in X}$ then necessarily ${\displaystyle x\in S.}$ The space ${\displaystyle (X,\tau )}$ is called a Fréchet–Urysohn space if ${\displaystyle \operatorname {scl} _{X}S~=~\operatorname {cl} _{X}S}$ for every subset ${\displaystyle S\subseteq X,}$ which happens if and only if every subspace of ${\displaystyle (X,\tau )}$ is a sequential space. Every first-countable space is a Fréchet–Urysohn space and thus also a sequential space. All pseudometrizable spaces, metrizable spaces, and second-countable spaces are first-countable.

A sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ in a set ${\displaystyle X}$ is by definition a function ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$ whose value at ${\displaystyle i\in \mathbb {N} }$ is denoted by ${\displaystyle x_{i}}$ (although the usual notation used with functions, such as parentheses ${\displaystyle x_{\bullet }(i)}$ or composition ${\displaystyle f\circ x_{\bullet },}$ might be used in certain situations to improve readability). Statements such as "the sequence ${\displaystyle x_{\bullet }}$ is injective" or "the image (i.e. range) ${\displaystyle \operatorname {Im} x_{\bullet }}$ of a sequence ${\displaystyle x_{\bullet }}$ is infinite" as well as other terminology and notation that is defined for functions can thus be applied to sequences. A sequence ${\displaystyle s_{\bullet }}$ is said to be a subsequence of another sequence ${\displaystyle x_{\bullet }}$ if there exists a strictly increasing map ${\displaystyle l_{\bullet }:\mathbb {N} \to \mathbb {N} }$ (possibly denoted by ${\displaystyle l_{\bullet }=\left(l_{k}\right)_{k=1}^{\infty }}$ instead) such that ${\displaystyle s_{k}=x_{l_{k}}}$ for every ${\displaystyle k\in \mathbb {N} ,}$ where this condition can be expressed in terms of function composition ${\displaystyle \circ }$ as: ${\displaystyle s_{\bullet }=x_{\bullet }\circ l_{\bullet }.}$ As usual, if ${\displaystyle x_{l_{\bullet }}=\left(x_{l_{k}}\right)_{k=1}^{\infty }}$ is declared to be (such as by definition) a subsequence of ${\displaystyle x_{\bullet }}$ then it should immediately be assumed that ${\displaystyle l_{\bullet }:\mathbb {N} \to \mathbb {N} }$ is strictly increasing. The notation ${\displaystyle x_{\bullet }\subseteq S}$ and ${\displaystyle \operatorname {Im} x_{\bullet }\subseteq S}$ mean that the sequence ${\displaystyle x_{\bullet }}$ is valued in the set ${\displaystyle S.}$

The function ${\displaystyle f:X\to Y}$ is called a sequence covering if for every convergent sequence ${\displaystyle y_{\bullet }}$ in ${\displaystyle Y,}$ there exists a sequence ${\displaystyle x_{\bullet }\subseteq X}$ such that ${\displaystyle y_{\bullet }=f\circ x_{\bullet }.}$ It is called a 1-sequence covering if for every ${\displaystyle y\in Y}$ there exists some ${\displaystyle x\in f^{-1}(y)}$ such that every sequence ${\displaystyle y_{\bullet }\subseteq Y}$ that converges to ${\displaystyle y}$ in ${\displaystyle (Y,\sigma ),}$ there exists a sequence ${\displaystyle x_{\bullet }\subseteq X}$ such that ${\displaystyle y_{\bullet }=f\circ x_{\bullet }}$ and ${\displaystyle x_{\bullet }}$ converges to ${\displaystyle x}$ in ${\displaystyle (X,\tau ).}$ It is a 2-sequence covering if ${\displaystyle f:X\to Y}$ is surjective and also for every ${\displaystyle y\in Y}$ and every ${\displaystyle x\in f^{-1}(y),}$ every sequence ${\displaystyle y_{\bullet }\subseteq Y}$ and converges to ${\displaystyle y}$ in ${\displaystyle (Y,\sigma ),}$ there exists a sequence ${\displaystyle x_{\bullet }\subseteq X}$ such that ${\displaystyle y_{\bullet }=f\circ x_{\bullet }}$ and ${\displaystyle x_{\bullet }}$ converges to ${\displaystyle x}$ in ${\displaystyle (X,\tau ).}$ A map ${\displaystyle f:X\to Y}$ is a compact covering if for every compact ${\displaystyle K\subseteq Y}$ there exists some compact subset ${\displaystyle C\subseteq X}$ such that ${\displaystyle f(C)=K.}$

In analogy with the definition of sequential continuity, a map ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is called a sequentially quotient map if

${\displaystyle f:(X,\operatorname {SeqOpen} (X,\tau ))\to (Y,\operatorname {SeqOpen} (Y,\sigma ))}$

is a quotient map,[5] which happens if and only if for any subset ${\displaystyle S\subseteq Y,}$ ${\displaystyle S}$ is sequentially open ${\displaystyle (Y,\sigma )}$ if and only if this is true of ${\displaystyle f^{-1}(S)}$ in ${\displaystyle (X,\tau ).}$ Sequentially quotient maps were introduced in Boone & Siwiec 1976 who defined them as above.[5]

Every sequentially quotient map is necessarily surjective and sequentially continuous although they may fail to be continuous. If ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is a sequentially continuous surjection whose domain ${\displaystyle (X,\tau )}$ is a sequential space, then ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is a quotient map if and only if ${\displaystyle (Y,\sigma )}$ is a sequential space and ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is a sequentially quotient map.

Call a space ${\displaystyle (Y,\sigma )}$ sequentially Hausdorff if ${\displaystyle (Y,\operatorname {SeqOpen} (Y,\sigma ))}$ is a Hausdorff space.[6] In an analogous manner, a "sequential version" of every other separation axiom can be defined in terms of whether or not the space ${\displaystyle (Y,\operatorname {SeqOpen} (Y,\sigma ))}$ possess it. Every Hausdorff space is necessarily sequentially Hausdorff. A sequential space is Hausdorff if and only if it is sequentially Hausdorff.

If ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is a sequentially continuous surjection then assuming that ${\displaystyle (Y,\sigma )}$ is sequentially Hausdorff, the following are equivalent:

1. ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is sequentially quotient.
2. Whenever ${\displaystyle y_{\bullet }\to y}$ is a convergent sequence in ${\displaystyle Y}$ then there exists a convergent sequence ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle X}$ such that ${\displaystyle f(x)=y}$ and ${\displaystyle f\circ x_{\bullet }}$ is a subsequence of ${\displaystyle y_{\bullet }.}$
3. Whenever ${\displaystyle y_{\bullet }}$ is a convergent sequence in ${\displaystyle Y}$ then there exists a convergent sequence ${\displaystyle x_{\bullet }}$ in ${\displaystyle X}$ such that ${\displaystyle f\circ x_{\bullet }}$ is a subsequence of ${\displaystyle y_{\bullet }.}$
   • This statement differs from (2) above only in that there are no requirements placed on the limits of the sequences (which becomes an important difference only when ${\displaystyle Y}$ is not sequentially Hausdorff).
   • If ${\displaystyle f:X\to Y}$ is a continuous surjection onto a sequentially compact space ${\displaystyle Y}$ then this condition holds even if ${\displaystyle Y}$ is not sequentially Hausdorff.

If the assumption that ${\displaystyle Y}$ is sequentially Hausdorff were to be removed, then statement (2) would still imply the other two statement but the above characterization would no longer be guaranteed to hold (however, if points in the codomain were required to be sequentially closed then any sequentially quotient map would necessarily satisfy condition (3)). This remains true even if the sequential continuity requirement on ${\displaystyle f:X\to Y}$ was strengthened to require (ordinary) continuity. Instead of using the original definition, some authors define "sequentially quotient map" to mean a continuous surjection that satisfies condition (2) or alternatively, condition (3). If the codomain is sequentially Hausdorff then these definitions differs from the original only in the added requirement of continuity (rather than merely requiring sequential continuity).

The map ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is called presequential if for every convergent sequence ${\displaystyle y_{\bullet }\to y}$ in ${\displaystyle (Y,\sigma )}$ such that ${\displaystyle y_{\bullet }}$ is not eventually equal to ${\displaystyle y,}$ the set ${\displaystyle \bigcup _{\stackrel {i\in \mathbb {N} ,}{y_{i}\neq y}}f^{-1}\left(y_{i}\right)}$ is not sequentially closed in ${\displaystyle (X,\tau ),}$[5] where this set may also be described as:

${\displaystyle \bigcup _{\stackrel {i\in \mathbb {N} ,}{y_{i}\neq y}}f^{-1}\left(y_{i}\right)~=~f^{-1}\left(\left(\operatorname {Im} y_{\bullet }\right)\setminus \{y\}\right)~=~f^{-1}\left(\operatorname {Im} y_{\bullet }\right)\setminus f^{-1}(y)}$

Equivalently, ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is presequential if and only if for every convergent sequence ${\displaystyle y_{\bullet }\to y}$ in ${\displaystyle (Y,\sigma )}$ such that ${\displaystyle y_{\bullet }\subseteq Y\setminus \{y\},}$ the set ${\displaystyle f^{-1}\left(\operatorname {Im} y_{\bullet }\right)}$ is not sequentially closed in ${\displaystyle (X,\tau ).}$

A surjective map ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ between Hausdorff spaces is sequentially quotient if and only if it is sequentially continuous and a presequential map.[5]