Definition of local connectedness by Zorich's Mathematical Analysis II

Question

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!

Answer 1

Zorich's definition of "local connectedness" is strictly weaker than Munkres' definition.

It's trivial to show that a space $X$ satisfying Munkres' definition also satisfies Zorich's definition, since $X$ is open in $X$, so there must be a connected (open) neighborhood $U \subseteq X$ of $x$ for each $x \in X$.

However, there are spaces which satisfy Zorich's definition but not Munkres' definition. An easy example is the topologist's sine curve $A = {\rm Cl} S$ (as a subspace of $\mathbb{R}^2$). Every point has a connected (open) neighborhood, namely, the entire space $A$ itself, but points in the vertical strip $0 \times [-1, 1]$ do not satisfy the property that every (open) neighborhood of $x$ contains a connected (open) neighborhood of $x$.

The definition given by Munkres is more widely adopted, because it captures the "local" nature of "local connectedness", in the sense that in such spaces, we can find arbitrarily small neighborhoods about points with a specific property (here, connectedness).

Small note: it is not necessary to stipulate that a "neighborhood" is a "non-empty open set containing a point", since, clearly, it must contain at least that point.

Answer 2

This is just a supplement to K. Jiang's answer giving an example that Zorich's definition does not agree with the standard definition of local connectedness (given by Munkres and wikipedia).

Actually we can say that Zorich's definition does not make sense.

1. In the sense of Zorich each connected space is locally connected.

2. In exercise 9.4.1 / 4. b) he claims that the closed topologist's sine curve $E$ is connected but not locally connected. It is in fact connected, but then it contradicts 1. Moreover, as K. Jiang correctly states, $E$ is not locally connected in the standard interpretation.

Conclusion: Forget Zorich's definition. He made a mistake.