Answer to Question #217087 in Differential Geometry | Topology for Prathibha Rose

Question

Give an example of a first countable space which is not second countable ,substantiate your claim

Answer 1

In toplogy a branch of mathematics ,a first countable space is a topology space satisfying the first axioms of countability each point has a countable neighborhood basis.That is ,for each point x in X there exists three sequence $\Nu$₁,$\Nu$₂,....of neighborhoods of x such that for $\Nu$ of x there exists an integer i with $\Nu$_i contained in $\Nu$. since every neighborhood of any point contains an open neighborhood of that point the neighborhood can be choosen without loss of generality to consist open neighborhoods.

Example

we must prove this is a local basis for y in Y so take an open set U' in Y containing y we must have that U' $=$U$\cap$ Y such that U is an open set in X containing y therefore there exists B$\in$B such that y$\in$B and B$\subsetneq$U .We conclude that y$\in$B$\cap$ Y $\subsetneq$ U' and clearly B$\cap$ U$\in$B' therefore B' is local basis for y in Y since we can find a countable basis for each point in we conclude Y is first countable