Page 124 - DMTH503_TOPOLOGY

Topology

Notes Theorem 3: In a topological space X each connected sub space of X is contained in a component
of X.
Proof: Let E be any connected subspace of X.

If E   then E is contained in every component of X.
,
Let E   and let x E
,
Then x  X
Let E be the union of all connected subsets of X containing x. Then, E is a component of X
x x
containing x.
Now, E is a connected set containing x and E is the largest connected set containing x. So E  E .
x x
Theorem 4: In a topological space (X, T), a connected subspace of X which is both open and closed,
in a component of X.

Proof: Let G be a connected subspace of X which is both open and closed.
If G = , then G is contained in every component.
If G , then G contains a point x  X and so
1
G  C (X,x ) = C
i
We shall show that G = C

In order to show that G = C, let us assume that G is a proper subset of C, so that
G  C   and G  C   where G  X C.

Since G is both open and closed, G is also both open and closed.


Also (G  C)  (G  C) = (G  G ) 
=   C  
and (G  C)  (G  C) = (G  G )  C  X  C  X

which shows that C is disconnected, which is a contradiction of the given fact that C is connected
Hence G = C.

Theorem 5: The product of any non-empty class of connected topological spaces is connected i.e.
connectedness is a product invariant property.

Proof: Let {x } be a non-empty connected topological spaces and X   X i be the product space.
i i
Let a  a   X and E be a component of a.
i
We claim that X  E E ( E is closed)

Let x   x  be any point of X and let G   {X : i  i ,...i }XG X....XG i
1
i
m
1
i
be any basic open set containing x.

Now H   {a } ;i i ,i ,....,i 1 2 m  X i 1  X  .... X i m is homeomorphic to X  X  ...X i m and is
i
2
i 1
i a
therefore connected
( connectedness is a topological property)
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