cmi-entrance 2020 Q13

cmi-entrance · India · pgmath 10 marks Proof Proof That a Map Has a Specific Property

Let $F \subseteq \mathbb{R}^{3}$ be a non-empty finite set, and $X = \mathbb{R}^{3} \backslash F$, taken with the subspace topology of $\mathbb{R}^{3}$. Show that $X$ is homeomorphic to a complete metric space. (Hint: Look for a suitable continuous function from $X$ to $\mathbb{R}$.)

Let $F \subseteq \mathbb{R}^{3}$ be a non-empty finite set, and $X = \mathbb{R}^{3} \backslash F$, taken with the subspace topology of $\mathbb{R}^{3}$. Show that $X$ is homeomorphic to a complete metric space. (Hint: Look for a suitable continuous function from $X$ to $\mathbb{R}$.)

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