Does a countable fundamental neighborhood system always form a total order set?

Question

A point $x$ in a topology space $X$ has a countable fundamental neighborhood system. Does the countable fundamental neighborhood system necessarily has a total order under inclusion?

Answer 1

Here is a countable neighborhood base for the point $0$ in the space $\mathbb R$ which is not totally ordered by inclusion:

$$\{(-1.5,1.0),(-0.1,1.5),(-0.15,0.1),(-0.01,0.15),(-0.015,0.01),(0.001,0.015),\cdots\}$$

On the other hand, if $$\{N_1,\ N_2,\ N_3,\ N_4,\ N_5,\cdots$$ is any countable neighborhood base for a point $x$ in a topological space $X,$ then $$N_1,\ N_1\cap N_2,\ N_1\cap N_2\cap N_3,\ N_1\cap N_2\cap N_3\cap N_4,\ N_1\cap N_2\cap N_3\cap N_4\cap N_5,\dots$$ is a countable neighborhood base for $x$ which is totally ordered by inclusion.