Subspaces (a) Let (X, τ) be a topological space and A ⊆ X....

Subspaces (a) Let $(X, \tau)$ be a topological space and $A \subseteq X$. Prove that 1. If $F \subseteq A$ is closed in $A$, then $F = K \cap A$ for some closed $K$ in $X$. 2. If $B \subseteq A$, then $A \cap B^o \subseteq Int_A(B)$. (b) Let $(X, \tau)$ be a topological space and $F$ be a closed subset of $X$. Prove that a set $K \subseteq F$ is closed in $F$ iff it is closed in $X$.

(1) Prove that, in a topological space X, if U is open and C is closed, then U - C is open and C - U is closed. (2) Let U and E be open and closed subsets of a topological space X, respectively. (a) Prove that a subset A of U is relatively open in U if and only if it is an open subset of X. (b) Prove that a subset A of E is relatively closed in E if and only if it is a closed subset of X. (3) Prove that all half-open intervals in R (finite or infinite) are homeomorphic. (4) Let f : X → Y be a function between two topological spaces. (a) Prove that f is a continuous function if and only if f^-1(E) is a closed subset of Y for each closed subset E of Y. (b) Prove that f is a continuous function if and only if f(Ā) ⊂ f(A) for each subset A of X. (c) Suppose that f is a homeomorphism. Let a third topological space Z and a function g : Y → Z be given. Prove that g is continuous if and only if g ∘ f is continuous.

Sri K.

Question

Please give Ace some feedback