Is the union of finite and countably infinite sequence over alphabet $\Sigma=\{1\}$, countably infinite as well?
I understand this is similar a question to the one of finite and countably infinite strings over $\{0,1\}$, however I think that the previous argument is not valid in this case.
Now, countably infinite strings over $\{0,1\}$ can be of any kind, e.g.
$$e_i \in \Sigma^*, e_i=100000...$$ $$e_k \in \Sigma^*, e_k=110000...$$
However, I noticed that if I pose the same question with an alphabet of a character only, say $\{1\}$, there is only 1 infinite sequence that I can generate $11.....$, hence:
$$S = \Sigma^* \cup \{111..\}$$
From this assumption, I could map $0$ to the infinite $11...$ and assume that this set S is uncountable. Is this a valid argument? Can someone help me formalize this?