Connected neibourhood and path connected neibourhood

Question

I studying locally connectedness and locally path connectedness from MUNKRES but I can't understand the term connected neibourhood and path connected neibourhood.

I will be thankful ,if anyone explain it to me.

Answer 1

If I slip on this one, I’m sure others will correct me.

A neighborhood of a point $P$ is just an open set containing $P$. A set $S$ of any kind is connected if it is not the disjoint union of two nonempty open subsets. A set $S$ is path connected if for any two points, $s,t\in S$, there is a continuous map $\gamma:[0,1]\to S$ with $\gamma(0)=s$ and $\gamma(1)=t$. In plain language, if you can draw a curve within $S$ from $s$ to $t$. It should be not hard to show that path connected implies connected.

Now, as I recall, in $\Bbb R^n$, a connected open set is automatically path connected. (This is the one piece of information I’m not totally sure of.) So you’re not going to find a connected open set in $\Bbb R^n$ that isn’t path connected.

Two examples in $\Bbb R^2$, the ordinary Euclidean plane:

The open unit disc centered at the origin is connected and path connected. And it’s a neighborhood of the origin.

If you take the union of the open discs about the origin and about $(0,5)$, both of radius $1$, you get a neighborhood of the origin that’s not connected. And certainly not path connected, ’cause within that set there’s no path from the one center to the other.

For an example of a topological space that’s connected but not path connected, you have to go weird: take the graph of the function $f:\langle0,1]\to\Bbb R$ by the rule $f(x)=\sin(1/x)$. I’m sure you see that this graph is contained in the strip of width $2$ centered on the $x$-axis, with infinitely many wiggles, getting closer and closer together as you look more and more closely near the $y$-axis. Call this graph $G$. It’s not our topological space; rather our space $X$ is $G\cup I$, where $I$ is the $y$-axis. It takes a bit of work to show that this space $X$ is connected, but not path connected; and since it’s open in itself, it’s a neighborhood of any of its points.