I was looking for examples of spaces that are not first countable.  they give the example of $[0,\omega_1]$. I think I understand that $\omega_1$ is  a limit point of $[0,\omega_1)$ since if $(\alpha,\beta)$ is an open containing $\omega_1$ it contains an ordinal smaller than $\omega_1$ and thus this ordinal is in $[0,\omega_1)$. Now why is there no sequence of $[0,\omega_1)$ converging to $\omega_1$. It is not entirely clear to me how to prove this. Also I don't really understand why this implies that $\omega_1$ doesn't have a countable basis.
The topological space $X=\omega_1\cup \{\omega_1\}$ is not first countable but $\omega_1$ is. Here $\omega_1$ denotes the set of all countable ordinals and $\{\omega_1\}$ is the first uncountable ordinal. Cardinality of $\omega_1$ is $\aleph_1.$And also $\omega_1$ is called the limit ordinal meaning it is the limit point of the set $\omega_1.$  Here I want to prove two things. $1)$ $\omega_1$ is first countable. and $2)$ $\{\omega_1\}$ is the limit point of $\omega_1$. But since no sequence converges to $\{\omega_1\}$, using this I could say that our space $X$ is not first countable.  $1)$ The proof of this I got from  with just one question: it says we need to prove that such an $S(y)$ exists,and I'm thinking $S(y)=\cap\{u\in \omega_1:u\gt y\}.$ This is correct$?$ But here is a chance that the countable base around $y$ may be finite and not countably infinite.  $2)$ If possible let us assume that $\omega_1$ is not a limit point of the set $\omega_1.$ $U$ be a nbd of $\omega_1$ that does not intersect $\omega_1.$ Then what are the elements of $U$ like? The ones coming before $\omega_1$  cannot be uncountable because $\omega_1$  is the first. Then they must be countable. This is absurd so no such $U$ exists. Thus $\{\omega_1\}$ is a limit point.  Are my proofs correct? Thank you.
In a script I'm trying to find the Terminal.app window containing a specific tty device and raise that window to the top. Here's what I have in Applescript so far:  tell application "System Events"   set foundWin to false   if (name of processes) contains "Terminal" then     tell application "Terminal"       set theTabs to first tab of every window where tty is "$(tty)"       repeat with theTab in theTabs         if class of theTab is tab then           set theWin to (first window whose tabs contains theTab)           set selected tab of theWin to theTab           set foundWin to true         end if       end repeat     end tell   end if   if foundWin then     --RAISE THE WINDOW?!   end if end tell   Where I'm getting stuck is the "raising the window" part.  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