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Recently I encountered the term 'cofinal segment' in the paper 'The Point of Continuity Property, Neighbourhood Assignments and Filter Convergences' by Ahmed Bouziad, example $2.3$.

Question: What is the definition of cofinal segment? If possible, can someone provide me an example so that I can understand it fully.

The following is the text:

Let $\omega_1$ denote the first uncountable ordinal. Let $X$ be the set of all $\phi \in \{ 0,1 \} ^{\omega_1}$ such that $\phi$ is constant on some cofinal segment $[\alpha, \omega_1), \alpha < \omega_1$.

Idonknow
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1 Answers1

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Let $\langle X,\le\rangle$ be a linear order. Each $x\in X$ defines a closed cofinal segment of $X$,

$$[x,\to)=\{y\in X:x\le y\}$$

and an open cofinal segment

$$(x,\to)=\{y\in X:x<y\}\;.$$

A subset $C$ of $X$ is cofinal if for each $x\in X$ there is a $y\in C$ such that $x\le y$. The term segment isn’t entirely standard, but it indicates that there are no ‘holes’ in the cofinal set: if $x\in C$, $y\in X$, and $x\le y$, then $y\in C$ as well.

In the paper that you’re reading the term cofinal segment is defined, and we can see that the author means what I called a closed cofinal segment: he tells you that by cofinal segment of $\omega_1$ he means a set of the form $[\alpha,\omega_1)=\{\beta<\omega_1:\alpha\le\beta\}$.

Brian M. Scott
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