Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 yY, then yX since YX , 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 :bB(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):bB} 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:bDB } and then f(f-1(G))=f{U{b:bD}} G=U{f(b):bD}. Thus any V-open set is expressible as a union of {f(b): bB} {f(b):bB} is accountable base for V there for Y is second countable . Theorem: the property of being a first countable space is hereditary property