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  "body": "\n## Proof\n\nWe prove each implication separately.\n\n1. Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \\colon X \\to Y$.\n2. Write $B_X(a, \\delta)$ and $B_Y(b, \\varepsilon)$ for the [[open balls|0DC3MPR7C9]] of $X$ and $Y$. By the symmetry of a metric, $x \\in B_X(a, \\delta)$ if and only if $d_X(x, a) \u003c \\delta$.\n3. Assume condition 1. Let $V$ be an [[open set|01ZQQDCERP]] of $Y$, and let $a$ be in the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$.\n4. Then $f(a) \\in V$. By the [[description of the open sets of a metric space|0A26X79RTB]], there is $\\varepsilon \u003e 0$ with $B_Y(f(a), \\varepsilon) \\subseteq V$.\n5. Condition 1 gives $\\delta \u003e 0$ such that $d_X(x, a) \u003c \\delta$ implies $d_Y(f(x), f(a)) \u003c \\varepsilon$.\n6. Let $x \\in B_X(a, \\delta)$. By step 2 and step 5, $d_Y(f(x), f(a)) \u003c \\varepsilon$.\n7. By symmetry, $f(x) \\in B_Y(f(a), \\varepsilon) \\subseteq V$, so $x \\in f^{-1}(V)$. Thus $B_X(a, \\delta) \\subseteq f^{-1}(V)$.\n8. Each point of $f^{-1}(V)$ has a ball about it inside $f^{-1}(V)$. By the description of open sets, $f^{-1}(V)$ is open. This is condition 2.\n9. Now assume condition 2. Let $a \\in X$ and $\\varepsilon \u003e 0$. Put $V = B_Y(f(a), \\varepsilon)$.\n10. By [[open balls are open|0B2305GNAZ]], $V$ is open. By condition 2, $f^{-1}(V)$ is open in $X$.\n11. Since $d_Y(f(a), f(a)) = 0 \u003c \\varepsilon$, we have $f(a) \\in V$, so $a \\in f^{-1}(V)$.\n12. By the description of open sets, there is $\\delta \u003e 0$ with $B_X(a, \\delta) \\subseteq f^{-1}(V)$.\n13. Let $x \\in X$ with $d_X(x, a) \u003c \\delta$. By step 2, $x \\in B_X(a, \\delta)$, so $f(x) \\in V$.\n14. So $d_Y(f(a), f(x)) \u003c \\varepsilon$, and by symmetry $d_Y(f(x), f(a)) \u003c \\varepsilon$. This is condition 1.\n\n## Idea\n\nAn open set contains a ball about each of its points, and balls are open. Condition 1 says that a small ball about $a$ maps into a given ball about $f(a)$. Translating between balls and open sets gives each direction.\n",
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