X be a topology space show that H is open iff no limit point of His in H. 5/a/ Let (X,T) be a first countable space show that there exists a monotone decreasing local base at every point of X. of first countable space is hereditary property.

Show that if a Banach space X is non-trivial, X + {0}, 3). then X' + {0}.

Prove or disprove the following statement: A) Every T'3-space is T4-space?( explain your answer)

Find two examples for equivalet norms on three dimensional real space R³. .

I Prove that homotopy with respect to a domain 2 is a transitive property.

Please provide a clear and detailed solution

Q/Let X+6 and (Y₁Ty) topological space and het fox-cy function. if Tx is defined as [x = {f(G) | Ge Ty} then (Txofological Space

2/1f (xit) is top sPacest X+y there are two disjoint open sets containing them Prove {y} y ex isclosed set

Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

theoremo Let (XT) be topological space and het DEX then the following statements are equivalent Dis a dense Setin x if f is any closed Set of X Such that D≤f then F=X 3 if Bx is a base of x Space, then UND + $$UB D² = Q H-W// Suppose and Proof 4 Without & [2] through Point 131 then suppose [77 and Proof I De_ext (D) exterior -C