Given a topological space $X,$ there is a function $f_X : X \rightarrow \mathrm{Cardinals}$ defined as follows. Given a point $x \in X$, $f_X(x)$ is the cofinality of the poset whose elements are the neighbourhoods of $x,$ whose order relation is reverse-inclusion. (So $U \leq V$ iff $U \supseteq V$).
For example, to say that $X$ is first-countable is just to say that $f_X(x) \leq \aleph_0$ for all $x \in X$.
Specific question. What is an example of a connected topological space $X$ where $f_X$ is not constant?
General question. What is known about the possible behaviors of $f_X$ when $X$ is connected?
This cardinal function is better known as the character of a topological space at a point, and is denoted $\chi$.
For an example of a connected space where $\chi$ is not constant, consider the "extended long ray" $X = ( \omega_1 \times [0,1) ) \cup \{ \langle \omega_1 , 0 \rangle \}$ with the (lexicographic) order topology. It is pretty clear that $\chi ( X ; \langle \alpha , x \rangle ) = \aleph_0$ for all $\langle \alpha , x \rangle \in \omega_1 \times [0,1)$, however $\chi ( X ; \langle \omega_1 , 0 \rangle ) = \aleph_1$.
(I am unaware of any restrictions on this cardinal function in the class of connected spaces. Well, I guess $\chi ( X ; x )$ cannot be finite if $X$ is a connected Hausdorff space of size $>1$, but that's not really saying much.)