cmi-entrance 2019 Q3

cmi-entrance · India · pgmath 4 marks Proof True/False Justification
Let $(X, d)$ and $(Y, \rho)$ be metric spaces and $f : X \longrightarrow Y$ a homeomorphism. Choose the correct statement(s) from below:
(A) If $B \subseteq Y$ is compact, then $f^{-1}(B)$ is compact;
(B) If $B \subseteq Y$ is bounded, then $f^{-1}(B)$ is bounded;
(C) If $B \subseteq Y$ is connected, then $f^{-1}(B)$ is connected;
(D) If $\{y_n\}$ is Cauchy in $Y$, then $\{f^{-1}(y_n)\}$ is Cauchy in $X$.
Let $(X, d)$ and $(Y, \rho)$ be metric spaces and $f : X \longrightarrow Y$ a homeomorphism. Choose the correct statement(s) from below:\\
(A) If $B \subseteq Y$ is compact, then $f^{-1}(B)$ is compact;\\
(B) If $B \subseteq Y$ is bounded, then $f^{-1}(B)$ is bounded;\\
(C) If $B \subseteq Y$ is connected, then $f^{-1}(B)$ is connected;\\
(D) If $\{y_n\}$ is Cauchy in $Y$, then $\{f^{-1}(y_n)\}$ is Cauchy in $X$.