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Here (in the context of Abstract Elementary Classes) on the page 43 at the bottom,-6th line, what does it technically mean $$\leq_{\frak K_\lambda}-\text{increasing continuous}$$

? I think that this should be a condition on limit ordinals, but in his text, Shelah uses $\alpha$ for both, limit and successors ordinals (see the page 67 in the link above) and he writes in that -6th line

[...for] $\alpha<\lambda^+$.

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Note: the relevant definition is at the bottom of page $43$.

You aren't misunderstanding anything, but you are overthinking a bit. Shelah could have indeed written "for all limit $\alpha<\lambda^+$," as you observe, but he didn't need to: continuity is a vacuous condition at successor ordinals. Thinking about it topologically, each successor ordinal $\beta$ is an isolated point, and so there are no restrictions at all on how a continuous function needs to behave at $\beta$.

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  • $\begingroup$ You always respond kindly to my confusions. Thank you. Could you please also explain more closely the very last sentence from your answer? :each successor ordinal $β$ is an isolated point, and so there are no restrictions at all on how a continuous function needs to behave at $β$. $\endgroup$
    – user122424
    Dec 17, 2018 at 15:35
  • $\begingroup$ @user122424 Think about how any map from a discrete space is continuous; this is a more general description of the same thing. If $X$ is a topological space, $a\in X$ is an isolated point, and $f:X\rightarrow Y$ is a continuous map, then any $g:X\rightarrow Y$ which agrees with $f$ except possibly at $a$ is also continuous. Basically, isolated points are topologically uninteresting (at least, considered individually - the set of isolated points in a given space might actually be an interesting object). In particular, a function from an ordinal which is "continuous at limits" is continuous. $\endgroup$
    – Noah Schweber
    Dec 17, 2018 at 16:02
  • $\begingroup$ This is really only a useful comment if you're more familiar with point-set topology than set theory; if you're not, ignore it, it's only intended as intuitive motivation. $\endgroup$
    – Noah Schweber
    Dec 17, 2018 at 16:02

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