Download a first countable space is a topological space in which there exist a

Definition: a first countable space is a topological space in
which there exist a countable local base at each of its point.
Example : every discrete space is first countable space.
Example : (R,U) is first axiom space.
Theorem: the property of being a first countable space is
hereditary property;
Proof: let (X,T) is first countable topological space and let Y be
asubspace of X we want to prove that Y is first countable space;
Let yY, then yX since YX , since X is first countable space
there exist a countable T-local base B(y).now it is easy to see
that By(y)={Y∩b :bB(y)} forms a countable Ty-local base at y
hence Y is first countable space.
Theorem: the property of being a first countable space is a
topological property
Proof: H.W
Definition: a topological space (X,T) is said to be second
countable space iff there exist a countable base for T.
Example : (R,U) is second countable space.
Example : (R,D) is not second countable space.
Theorem: every second countable space is first countable space
Theorem: the property of being a second countable space is a
topological property
Proof: let f(X,T)→(Y,V) be a homeomorphism of asecond
countable space X into atopological space Y .
Let B be a countable base for T. we want to show that
{f(b):bB} is countable base for V .
The above collection is countable and since f is open map ,f(b)
is V-open subset of Y .
f-1(G) is T-open set since f is continuous and since B is abase
for T we have
f-1(G)=U{b:bDB } and then f(f-1(G))=f{U{b:bD}}
G=U{f(b):bD}.
Thus any V-open set is expressible as a union of {f(b): bB}
{f(b):bB} is accountable base for V there for Y is second
countable .
Theorem: the property of being a first countable space is
hereditary property