In: Advanced Math
Let X and Y be T2-space. Prove that X*Y is also T2
Given
and
be
- spaces .
To prove that
is also a
- spaces suppose
and
be two elements of
.
and
.
As
and
is a
- spaces so there exist two open set
and
such that
,
with
.
As
and
is a
- spaces so there exist two open set
and
such that
,
with
.
Now since
,
is open in
and
,
is open in
so
and
is open in
.
also
as
,
.
and
.
So for all
and
in
there exist two open set
and
in
such that
and
with
.
Hence
is also a
- spaces .
.
.
.
.
If you have any doubt or need more clarification at any step please comment .