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SSC Pure Mathematics QUESTION #11173
Question 1
If $Y$ is a subset of $(X,d)$, then:
  • Every open set in $Y$ is open in $X$
  • Every open set in $X$ is open in $Y$
  • $O$ is open in $Y \Leftrightarrow O$ is open in $X$
  • $O$ is open in $Y \Leftrightarrow O = Y\cap G$ where $G$ is open in $X$✔️
Correct Answer Explanation
In a metric subspace $(Y,d)$ of $(X,d)$, a set $O\subseteq Y$ is open in $Y$ if and only if $O = Y\cap G$ for some open set $G$ in $X$ (subspace topology). This is the subspace topology definition. Open sets in $X$ need not be open in $Y$, and open sets in $Y$ need not be open in $X$.