I am trying to solve miscellaneous exercise 6 in Chapter 2 of Artin's book, *Algebra*. Below is the statement of the problem.

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> Let $a = (a\_1, \ldots, a\_k)$ and $b = (b\_1, \ldots, b\_k)$ be points in $k$-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0,1]$ with values in $\mathbb{R}^k$, a function $X: [0,1] \to \mathbb{R}^k$, sending $t \rightsquigarrow X(t) = (x\_1 (t), \ldots, x\_k (t))$, such that $X(0) = a$ and $X(1) = b$. If $S$ is a subset of $\mathbb{R}^k$ and if $a$ and $b$ are in $S$, define $a \sim b$ if and $a$ and $b$ can be joined by a path lying entirely in $S$.
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> (a) Show that $\sim$ is an equivalence relation on $S$. Be careful to check that any paths you construct stay within the set $S$.
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> (b) A subset $S$ is path connected if $a \sim b$ for any two points $a$ and $b$ in $S$. Show that every subset $S$ is partitioned into path-connected subsets with the property that two points in different subsets cannot be connected by a path in $S$.
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> (c) Which of the following loci in $\mathbb{R}^2$ are path-connected: $\{x^2 + y^2 = 1\}$, $\{xy = 0\}$, $\{xy = 1\}$?
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I'm fine with parts (a) and (b). My questions and uncertainties are:

1. Artin doesn't give a metric on $\mathbb{R}^k$, but I assume when he says continuous, he means with respect to the standard Euclidean metric on $\mathbb{R}^k$. The metrics on $\mathbb{R}^k$, I believe, are all equivalent in the sense that they induce the same notion of continuity, so I believe it is fine to use the Euclidean metric. Some confirmation on this would be appreciated.
2. I'm a bit stumped on part (c). The first thing I did was plot each of the loci, but I don't have any intuition for which of these are path-connected from the plot or where the interval $[0,1]$ comes into play.

I'd appreciate some help on how to appropriate this problem and how to think about path-connectedness geometrically.