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I notice the term path connected neighborhood. In my text,A First Course In Topology there is no mention of it

I looked on MSE and all over and could not find it .This term isn’t in my text when dealing with path connectedness and it’s components.

Here is my definition. It is pretty poor

Neighborhood

Let x $\in$ X A set A $\subset$ X is a nhbd of x if there is an open set U $\subset$ X s.t x $\in U \subset$ A

Definition of path connected set

A subset A of topological space X is path connected if any two points in A can be joined by path in A

Def. Path Connected Neighborhood

For x $\in U \subset A$ such that for a,b$\in $A there is continuous function (path) p:[0,1]$\mapsto$ A s.t p(0)=a to p(1)=b

I would like a nice clear one that I can use

Thanx

  • A neighborhood (of a point) is just an open set (containing the point). A path-connected set is a set which contains a path between each pair of its points. – Gerry Myerson Nov 27 '21 at 00:56
  • So we have x,y $\in $X and A $\subset$ X,where p:[0,1]$\mapsto$ p(0)=x to p(1)=y and a open set U of X then x,y$\in$ U$\subset$ X –  Nov 27 '21 at 01:26

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In general, a neighbourhood of $x\in X$ in a topological space $(X,\tau)$ is a subset $N$ such that there is an open set $U\in \tau$ such that $x\in U$ and $U\subseteq N$.

A path-connected neighbourhood of $x$ is a neighbourhood $N$ of $x$ which is also a path-connected subset of $(X,\tau)$. Namely, it's a neighbourhood of $x$ which satisfies the property that for all $v,w\in N$ there is a continuous function $f:[0,1]\to X$ such that $f(0)=v$, $f(1)=w$ and $f(t)\in N$ for all $t\in[0,1]$.

Some authors like to make the semantical substution "neighbourhood" = "open neighbourhood" (i.e. a neighbourhood of $x$ which is also an open subset of $(X,\tau)$), but usually they give the proper definition of neighbourhood where it is relevant, and then they just specify that they'll be discussing only open neighbourhoods for a while.