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In each subpart below you are given a metric space $(X, d)$ and a subspace $A ⊆ X$. You are also given a subset $U ⊆ A$. In each case, is $U$ an open subset of $A$ in the subspace metric on $A$? Justify your answer fully.

(a) Let $(X, d)$ be $\mathbb{R}$ with the usual metric. Let $A$ be the half-open interval $[0, 4)$. Let $U = (0, 1)$.

(b) Let $(X, d) = (\mathbb{R}^2, D)$ where $D$ is the railway metric. Let $A = \{ (x, y) ∈ \mathbb{R}^2 \text{ s.t. } y = 1 \} $. Let $U = \{(0, 1)\}$, that is $U$ consists of a single point whose $X$-coordinate is $0$ and $Y$ -coordinate is $1$.

Please help, trying to do past exam paper questions.

G. Chiusole
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MathGeek1998
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    Prove or disprove that $U$ can be written as $A\cap V$ where $V$ is an open set in $X$. Add the results to your question and tell us where you got stuck. – drhab Mar 16 '20 at 14:02
  • What exactly is the "railway metric"? – Math1000 Mar 16 '20 at 16:14
  • Railway metric on X is $D(x,y)= d_2(x,y)$ if $x,(0,0),y$ are collinear , $D(x,y)=||x|| + ||y||$ if $x,(0,0),y$ are not collinear for all $x,y \in X$. – MathGeek1998 Mar 16 '20 at 18:05

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