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Definition:
A topological space X is said to be locally connected at a point x ∈ X iff for each
open neighborhood U of x there is a connected open neighborhood V of x contained
in U:
x∈V⊂U⊂X
We say that X is locally connected if it is locally connected at each of its points
Example:
X={a,b},T=X,Ø,{a},{b},}
Since every subset of discrete space is open and every singleton set is connected the x
is locally connected.
Clearly discrete space is disconnected
Theorem: If X is locally connected, then each component C of X is open in X.
Proof. Let x ∈ C. Then U = X is a neighborhood of x, so by local connectivity there is
a connected neighborhood V of x with V ⊂ U = X. Since V is connected, V ⊂ C.
Hence C contains a neighborhood around each of its points, and must be open.
Theorem: the image of a locally connected space which is both continuous and open is
locally connected.
Proof :H.W
Definition :. Given points x, y ∈ X a path in X from x to y is a map f : [a, b] → X
with f(a) = x and f(b) = y, where [a, b] ⊂ R.
a subset E of X is path connected if for any two points x and y of E there exists a path
in E from x to y.
Lemma . A path connected space is connected.
Proof : Let E be path connected subset of X if E=Ø then E is connected , thus E≠Ø
and suppose that U and V separate X. Choose points x ∈ U and y ∈ V , and a path f :
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[a, b] → X in X from x to y. Then f−1(U) and f−1(V ) form a separation of the
connected space [a, b], which is impossible.
Lemma . The continuous image of a path connected space is path connected.
Proof. If g : X → Y is a map, any two points in f(X) can be written as g(x) and g(y) for
x, y ∈ X. Since X is path connected, there is a path f : [a, b] → X in X from x to y.
Then g ◦ f : [a, b] → Y is a path in f(X) from f(x) to f(y). Hence f(X) is path
connected.
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