Download Locally connected and locally path connected spaces

Lecture notes /topology / lecturer : Zahir Dobeas AL-nafie
*Locally connected and locally path connected
spaces*
Introduction:
One of the basuic problems of topology is to determine whether
tow given toplogical spaces are homeomorphic or not .there is no
method for solving this problem in genral ,but technique do exist
that apply in particular casis .
Showing that two spaces are homeomorphic is matter of
constructing of continuous mapping from one to the other having
continuous inverse ,and constructing continuous functions is a
problem that we have developed technique to handle.
Showing that two spaces are homeomorphic is different
matter .for that one must show that continuous functions with
continuous inverse dose not exist ,if one can find some topological
property that holds for one space but not for the other ,then the
problem is solved ,the space can not be homeomorphic .the closed
interval [0,1] can be homeomorphic to the open interval (0,1),for
instance ,because the first space is copact but the second one is
not .and the real line R can not be homeomorphic to the long line L
because R has countable basis but the second one is not
Def:
A space X is locally path connected each point x if every
neighborhood of x contains a path connected neighborhood of x,
the space X is locally path connected if it is locally path connected
at each of its points.
*A space is locally path connected iff it has a basis of path
connected subsets, and consider the table
connected
Locally
connected
Yes
Yes
No
No
Ø
No
No
Proposition:The space X is locally path connected if and only if open
subsets have open path components (open in the open), In a
locally path connected space the path component are clopen .
(Closed and open).
Proof:Assume that X is locally connected and let U be an open
subset, consider a component C of U the claim is that C is open,
let x be a point in C choose a connected neighborhood V of x such
that V⊂U, since V is connected and interest V, V⊂C this shows
that C is open, in particular, the components (of the open set X)
are open and since they form partition, also closed.
Conversely
Assume that open sub sets have open components, let x be a
point of X and U neighborhood of x contained in U.
Theorem:In a locally path connected space, the path components
and the are the same.
Proof:Suppose that X is locally path connected and each path
component P is contained in unique component C since C is
connected and P is clopen
.