Long right-hand side

The long right-hand side is a topological Espace similar to the real Droite, “in much longer”.

Definition

As an ordered unit, the long line, $L$ , are the product of the first Ordinal indénombrable $\ omega_1$ and of the whole of positive or null realities, the order on the product being the collating Sequence (giving the most weight to the element of $\ omega_1$ ).

As a topological space, it is this ordered unit provided with the Topologie of the order (the open intervals form a base of topology). This topological space is a topological Variété on separable board not . Better, one can provide it with a smooth structure of differentiable Variété ( i.e. of class $C^ \ infty$ ), and even real Analytique ( $\ omega$ ).

Alternatives of the definition consist in withdrawing the origin, or indefinitely prolonging the line towards the left in the same way that towards the line. The term of “long right-hand side” can, according to the authors, to indicate any of these three spaces. We adopt convention here that there is an edge on the left.

Properties

For all $x$ in $L$ (long line in question), the interval closed is homeomorphic with the real interval . However, $L$ has exceptional properties. For example:

Any continuation increasing with values in $L$ has a limit (that rises almost immediately from the corresponding property for $\ omega_1$ , which is itself a simple consequence of a very weak form of the Axiome of the choice). In particular, all following values in $L$ admits a value of adherence (and, if she does not admit of it that one, converges towards this value); because all following values in $L$ is limited. It follows that any function continues $\ mathbb R$ towards $L$ is limited.
In a way perhaps more surprising, any continuous function of $L$ towards $\ mathbb R$ is limited. (Indeed, if it is not the case, if $f: L \ to \ mathbb R$ continuous is not limited, one finds $x_0$ in $L$ such as $f (x_0) >0$ , then $x_1$ in $L$ such as $f (x_1) >1$ and $x_1>x_0$ , then $x_2>x_1$ such as $f (x_2) >2$ , and so on. That is to say $x$ limit of the continuation $x_0, x_1, x_2, \ dots$ ; while applying the continuity of $f$ in $x$ , one arrives at a contradiction.) One has in fact well better: any continuous function of $L$ towards $\ mathbb R$ is constant starting from a certain point.
the Compactifié de Stone-Cech of $L$ is obtained by adding only one point with $L$ , ad infinitum on the right. It is thus also its Compactifié d' Alexandroff.
Any continuous application injective (thus strictly increasing) of $L$ in $L$ is not limited. Moreover, one such application has arbitrarily large fixed points (thus a noncountable infinity of fixed points).

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