Getting acquainted with open and closed sets

Task number: 3165

The open sphere with center \(x\), radius \(r\) in \(\mathbb R^2\), and the Eucledian metric is for the purpose of this problem denoted as \(B(x,r)\).

Consider the metric space \((X,\rho)\), where \(X = \{(0, 0)\} \cup B((0, 2), 1)\) is a subset of \(\mathbb R^2\) and \(\rho\) is the Euclidean metric restricted to the set \(X\).

Determine whether the following subsets \(X\) are open or closed or both.

  • Variant 1

    \(\{(x_1, x_2) \in X\colon x_1 = 0 \}\).

  • Variant 2

    \(\{(0, 0)\}\).

  • Variant 3

    \(B((0, 2), \frac12)\).

Difficulty level: Easy task (using definitions and simple reasoning)
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