Sequential space

In Mathematical, and more particularly in Topology and Theory of the numbers, a sequential space is a topological Espace which checks the axiom far from restrictive of denombrability.

These spaces constitute the most general class of Espace S for which the continuations entirely determine the Topologie. They have interesting properties in Théorie of the categories.

Definitions

That is to say X a topological Space.

a subset U of X is known as “sequentially open” if any continuation ( X _N) of X convergent towards a point of U converges in U

a subset F of X known as “is sequentially closed” if the convergence of a continuation ( X _N) of F towards X implies that X belongs to F .

Any subset open of X is sequentially open.

A sequential space is a Espace satisfying one of the following equivalent conditions:

Any subset sequentially open of X is open;
Any subset sequentially closed of X is closed.

History

Although spaces satisfying these conditions were studied, in an implicit way, well before the explicit formulation of sequential spaces, one allots the first rigorous definition of those to S.P. Franklin in 1965.

Are it tried to answer the question “Which the classes of topological spaces which can be definite completion knowing only their convergent continuations? ”. Franklin leads to the definition above.

Equivalent definitions

Oneself X a space, then the following properties are equivalent:

X is sequential;
X is the quotient of a countable space;
X is the quotient of a metric Espace;
For all topological Space Y and any application F : X → Y , F is continuous If and only if, for any continuation of points ( X _N) in X converges towards X , the continuation ( F ( X _N)) converge towards F ( X ).

Sequential closing

That is to say a subset $A \ subset X$ of a space $X$ , sequential closing $\ left_ {\ mbox {seq}}$ is the unit:

\ left_ {\ mbox {seq}} = \ {X \ in X: \ {a_n \} \ to X, a_n \ in has \}

i.e. the whole of all the points $x \ in X$ for which there exists a continuation of $A$ which converges towards $x$ . The Application:

$\ left \, \ cdot \, \ right_ {\ mbox {seq}}: With \ mapsto \ left_ {\ mbox {seq}}$

operator of sequential closing is called. He shares properties with the ordinary Fermeture S, in particular the fact that the empty set is sequentially closed:

$\ left \ right_ {\ mbox {seq}} = \ varnothing$

The sequentially closed units are always subsets of closed units:

$A \ subset \ left_ {\ mbox {seq}} \ subset \ overline {has}$

for all $A \ subset X$ — here $\ overline {has}$ indicates the ordinary closing of the $A$ unit.

Sequential closing commutates with the union:

$\ left B \ right_ {\ mbox {seq}} = \ left_ {\ mbox {seq}} \ cup \ left_ {\ mbox {seq}}$

for all $A, B \ subset X$ . However, contrary to ordinary closing, the operator of sequential closing is generally not idempotent, it can thus check:

$\ left \ left [\ right_ {\ mbox {seq}} \ right has] _ {\ mbox {seq}} \ \ left_ {\ mbox {seq}}$

and that, even when $A \ subset X$ is subset of a sequential space $X$ .

Space of Fréchet-Urysohn

The topological spaces for which sequential closing coincides with ordinary closing are called spaces of Fréchet-Urysohn, in according to Maurice Fréchet and Pavel Urysohn. They check:

$\ left_ {\ mbox {seq}} = \ overline {has}$

for all $A \ subset X$ . A space is of Fréchet-Urysohn If and only if each one of its subspaces is sequential.

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