More On Local Connectedness Versus Weak Local Connectedness
Theorem
Let X be a weakly locally connected space. Then X is locally connected.
Proof
It is sufficient to show that the components of open sets is open. Let U be open in X and let C be a component of U. Let x be an element of C. Then x is an element of U so that there is a connected subspace A of X contained in U and containing a neighbourhood V of x. Since A is connected and A contains x, A must be a subset of C (the component containing x). Therefore, the neighbourhood V of x is a subset of C. Since x was arbitrary, we have shown that each x in C has a neighbourhood V contained in C. This shows that C is open relative to U. Therefore, X is locally connected.
A certain infinite union of decreasing broom spaces is an example of a space which is weakly locally connected at a particular point, but not locally connected at that point.
Read more about this topic: Locally Connected Space
Famous quotes containing the words local and/or weak:
“The difference between de jure and de facto segregation is the difference open, forthright bigotry and the shamefaced kind that works through unwritten agreements between real estate dealers, school officials, and local politicians.”
—Shirley Chisholm (b. 1924)
“But God hath chosen the foolish things of the world to confound the wise; and God hath chosen the weak things of the world to confound the things which are mighty.”
—Bible: New Testament St. Paul, in 1 Corinthians, 1:27.