I am reading chapter 9,section 4,of Mathematical Analysis II, written by Zorich.
In Exercise 4 of the fourth section he defined the locally connectedness: a topological space $\left(X,\tau\right)$ is called locally connect at $x \in X$ if every $x \in X$ has a connected neighbourhood, and the definition of neighbourhood in his book means: a non-empty open set $U$ containing $x$.
And it is unlike definition from wiki said：a topological space $\left(X,\tau\right)$ is called locally connected at $x \in X$ if every open neighbourhood $U$ of $x$ contains an open and connected neighbourhood $V$ of $x$.Obviously，latter definition implies former definition.
Also, I read Topology by James Munkres, his definition of neighbourhood is as Zorich's. However, he defines local connectedness like this: a space $X$ is said to be locally connected at $x$ if for every neighborhood $U$ of $x$ there is a connected neighborhood $V$ of $x$ contained in $U$. And I had proved it is equivalent to wiki's definition.
Here is my question: I wonder if the definition by Zorich is correct. If so, how to prove it is equivalent to others definitions? Thanks!