nLab generalized sequential space

Idea

A generalisation of various kinds of spaces equipped with a relation between its elements and its sequences.

WARNING: “Generalised sequential space” is a placeholder name for a concept which may or may not have another name in the mathematics literature.

 Definition

A set $S$ is a generalised sequential space if it comes with a binary relation $x \to c$ between the set of all sequences in $S$ and $S$ itself, for $x \in S^\mathbb{N}$ and $c \in S$. A generalised sequential space is sequentially Hausdorff if the binary relation is a functional relation, and every sequentially Hausdorff generalised sequential space has a partial function

$$\lim_{n \to \infty} (-)_n:S^\mathbb{N} \to S$$

The morphisms between generalised sequential space are the sequential limit-preserving functions: a function $f$ between generalised sequential spaces $S$ and $T$ is sequential limit-preserving if for all sequences $x \in S^\mathbb{N}$ and elements $c \in S$, $x \to c$ implies that $(f \circ x) \to f(c)$, where $f \circ x \in T^\mathbb{N}$ is the sequence generated by precomposition of the sequence $x$ by the function $f$. The isomorphisms between generalised sequential space $S$ and $T$ are the bijections $f$ from $S$ to $T$ such that for all sequences $x \in S^\mathbb{N}$ and elements $c \in S$, $x \to c$ if and only if $(f \circ x) \to f(c)$.

Generalised net spaces

The above concept can also be generalised from sequences to nets.

WARNING: “Generalised net space” is a placeholder name for a concept which may or may not have another name in the mathematics literature.

A set $S$ is a generalised net space if for each directed set $I$, there is a binary relation $x \to_I c$ between the set of all $I$-indexed nets in $S$ and $S$ itself, for $x \in S^I$ and $c \in S$. A generalised net space is Hausdorff if each binary relation $x \to_I c$ is a functional relation for all directed sets $I$.

The morphisms between preconvergence spaces are the limit-preserving functions: a function $f$ between generalised net spaces $S$ and $T$ is limit-preserving if for all directed sets $I$, nets $x \in S^I$ and elements $c \in S$, $x \to_I c$ implies that $(f \circ x) \to_I f(c)$, where $f \circ x \in T^I$ is the net generated by precomposition of the net $x$ by the function $f$. These functions can be called netwise limit-preserving functions to contrast with sequential limit-preserving functions, which only preserve sequential limits in generalised net spaces. These functions can also be called continuous functions or pointwise continuous functions, since nets detect the continuity of functions between them in convergence spaces and topological spaces.

The isomorphisms between generalised net spaces $S$ and $T$ are the bijections $f$ from $S$ to $T$ such that for all directed sets $I$ and nets $x \in S^I$ and elements $c \in S$, $x \to_I c$ if and only if $(f \circ x) \to_I f(c)$.

 See also

• limit of a sequence
• sequentially Hausdorff space
• subsequential space
• generalised filter space
• metric space
• uniform space
• Cauchy space
• Booij premetric space