# $\leq_{\frak K_\lambda}$-increasing continuous

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

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 $$\beta$$.

• 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 $β$. Dec 17, 2018 at 15:35
• @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. Dec 17, 2018 at 16:02
• 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. Dec 17, 2018 at 16:02