# Definition of local connectedness by Zorich's Mathematical Analysis II

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!

• Most topologists would use the Wikipedia/Munkres definition. But maybe for Zorich his definition is adequate. Sep 23 at 19:01
• These definitions aren't the same Sep 23 at 19:02
• @MoisheKohan No, it is inadequate. See my answer. Sep 23 at 23:11
• It is clear that it is not equivalent to the standard one. The question is if it is adequate for Zorich's purposes or if some of his proofs fail because of the wrong definition. In order to decide, one has to read his book. Sep 24 at 1:00
• @MoisheKohan Zorich just mentioned locally connectedness as a exercise, and he didn't prove anything by using locally connectedness.
– MGIO
Sep 24 at 2:33

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.

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.

• It is clear that the definition given by Zorich is not equivalent to the standard one. The question is if it is adequate for Zorich's purposes or if some of his proofs fail because of the nonstandard definition. In order to decide, one has to read his book. Otherwise, one cannot say that Zorich has made a mistake. Sep 24 at 1:41
• @MoisheKohan The only occurence in the whole book is an exercise în an introductory chapter about topology. In this execise he claims that the closed topologist's sine curve is not locally connected. But due to his definition each connected space is locally connected. This shows that his definition does not decribe what he intended. I guess he simply "copied" the definition of locally compact which means that each point has a relatively compact neigborhood. One can also discuss whether this is a good definition for arbitrary spaces, but certainly it is fine for Hausdorff spaces. Sep 24 at 10:14