{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe describe the intervals of each stage by binary words, show that $\\varphi$ is a continuous bijection, and conclude by compactness.\n\n1. For a word $w = (w_1, \\ldots, w_n) \\in \\{0, 1\\}^n$, let $\\sigma(w) = \\sum_{k=1}^{n} 2 w_k / 3^k$ and $I_w = [\\sigma(w), \\sigma(w) + 3^{-n}]$. For $n = 0$, the empty word gives $I = [0, 1]$.\n2. Removing the open middle third of $I_w$ leaves $[\\sigma(w), \\sigma(w) + 3^{-n-1}] = I_{w0}$ and $[\\sigma(w) + 2 \\cdot 3^{-n-1}, \\sigma(w) + 3^{-n}] = I_{w1}$. Here $w0$ and $w1$ are $w$ followed by $0$ or $1$.\n3. By induction on $n$ and the construction of the [[Cantor set|0BXNDZFQME]], the closed intervals of $C_n$ are the $I_w$ with $w \\in \\{0, 1\\}^n$.\n4. By step 2, $I_{wv} \\subseteq I_w$ for each extension $wv$ of $w$.\n5. Distinct words $w \\neq w'$ of length $n$ give disjoint intervals. Let $k$ be the first index with $w_k \\neq w'_k$, say $w_k = 0$ and $w'_k = 1$, and let $u$ be the common first $k - 1$ letters.\n6. By step 4, $I_w \\subseteq I_{u0} = [\\sigma(u), \\sigma(u) + 3^{-k}]$ and $I_{w'} \\subseteq I_{u1} = [\\sigma(u) + 2 \\cdot 3^{-k}, \\sigma(u) + 3^{-k+1}]$. These two intervals are disjoint.\n7. For $a = (a_n)_{n \\ge 1} \\in \\{0, 1\\}^{\\mathbb{N}}$, write $a|n = (a_1, \\ldots, a_n)$. The series for $\\varphi(a)$ has nonnegative terms bounded by $2/3^k$, so it converges by the [[comparison test|04Z5G0THN6]] and the [[geometric series|01X5CVV1Q5]]. Also $\\sigma(a|n) \\le \\varphi(a) \\le \\sigma(a|n) + \\sum_{k \u003e n} 2/3^k = \\sigma(a|n) + 3^{-n}$.\n8. So $\\varphi(a) \\in I_{a|n} \\subseteq C_n$ for each $n$, and $\\varphi(a) \\in C$. Thus $\\varphi$ maps into $C$.\n9. $\\varphi$ is injective. If $a \\neq b$, let $n$ be an index with $a_n \\neq b_n$. Then $a|n \\neq b|n$, and $\\varphi(a) \\in I_{a|n}$, $\\varphi(b) \\in I_{b|n}$, which are disjoint by step 5.\n10. $\\varphi$ is surjective. Let $x \\in C$. For each $n$, $x \\in C_n$, so by steps 3 and 5 there is exactly one word $w^{(n)}$ of length $n$ with $x \\in I_{w^{(n)}}$.\n11. The first $n$ letters of $w^{(n+1)}$ form a word $v$ with $x \\in I_{w^{(n+1)}} \\subseteq I_v$, by step 4. By uniqueness, $v = w^{(n)}$.\n12. So there is one sequence $a$ with $a|n = w^{(n)}$ for all $n$. Then $x$ and $\\varphi(a)$ both lie in $I_{a|n}$, so $|x - \\varphi(a)| \\le 3^{-n}$ for all $n$. Hence $x = \\varphi(a)$.\n13. $\\varphi$ is continuous. Give $C$ the [[subspace topology|0D5YHQGDJY]]. Let $O \\subseteq \\mathbb{R}$ be open and $a \\in \\varphi^{-1}(C \\cap O)$. By [[the ball criterion|0A26X79RTB]], some $(\\varphi(a) - \\varepsilon, \\varphi(a) + \\varepsilon)$ lies in $O$.\n14. Choose $N$ with $3^{-N} \u003c \\varepsilon$. Let $W$ be the set of $b$ with $b_k = a_k$ for $1 \\le k \\le N$. Then $W = \\bigcap_{k=1}^{N} \\pi_k^{-1}(\\{a_k\\})$, where $\\pi_k(b) = b_k$ is the $k$-th projection.\n15. Each $\\{a_k\\}$ is open in the [[discrete topology|0S2TEHX4E1]], so $W$ is a finite intersection of open sets of the [[product topology|04S9HX3GAP]]. So $W$ is open, and $a \\in W$.\n16. For $b \\in W$, $\\varphi(a)$ and $\\varphi(b)$ both lie in $I_{a|N}$, so $|\\varphi(b) - \\varphi(a)| \\le 3^{-N} \u003c \\varepsilon$. So $W \\subseteq \\varphi^{-1}(C \\cap O)$.\n17. Thus $\\varphi^{-1}(C \\cap O)$ is a union of open sets, hence open, and $\\varphi$ is [[continuous|0SY9ECQ19Q]].\n18. The space $\\{0, 1\\}$ is [[compact|0QPHW703PV]]: from an open cover, one member containing $0$ and one containing $1$ form a finite subcover. By [[Tychonoff's theorem|0JTRSJKCFB]], $\\{0, 1\\}^{\\mathbb{N}}$ is compact.\n19. $C$ is [[Hausdorff|0MXB4CKHFW]]. For $x \\neq y$ in $C$, let $r = |x - y| / 2$. Then $C \\cap (x - r, x + r)$ and $C \\cap (y - r, y + r)$ are disjoint open sets of $C$ that contain $x$ and $y$.\n20. So $\\varphi$ is a continuous bijection from a compact space to a Hausdorff space. By [[this theorem|0430ETCMKD]], it is a [[homeomorphism|05GZ11PWRY]] onto $C$.\n\n## Idea\n\nThe binary word $a_1 \\ldots a_n$ records which of the $2^n$ intervals of the $n$-th stage contains $\\varphi(a)$. Distinct words give disjoint intervals, which makes $\\varphi$ a bijection, and agreeing on many first terms means being in the same tiny interval, which gives continuity. Compactness of the sequence space then makes the inverse continuous for free.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "54F65"
    ],
    "name": "Proof that the Cantor set is homeomorphic to the space of binary sequences",
    "proves": "0RQ8DHHR8V",
    "sources": [
      {
        "locator": "Examples",
        "url": "https://en.wikipedia.org/wiki/Cantor_space"
      }
    ],
    "tag": "542SG6445R",
    "uses": [
      "0BXNDZFQME",
      "0S2TEHX4E1",
      "04S9HX3GAP",
      "05GZ11PWRY",
      "0JTRSJKCFB",
      "0430ETCMKD",
      "0QPHW703PV",
      "0MXB4CKHFW",
      "0D5YHQGDJY",
      "0A26X79RTB",
      "0SY9ECQ19Q",
      "04Z5G0THN6",
      "01X5CVV1Q5"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/542SG6445R/",
  "links": [
    {
      "field": "proves",
      "target": "0RQ8DHHR8V"
    },
    {
      "field": "uses",
      "target": "0BXNDZFQME"
    },
    {
      "field": "uses",
      "target": "0S2TEHX4E1"
    },
    {
      "field": "uses",
      "target": "04S9HX3GAP"
    },
    {
      "field": "uses",
      "target": "05GZ11PWRY"
    },
    {
      "field": "uses",
      "target": "0JTRSJKCFB"
    },
    {
      "field": "uses",
      "target": "0430ETCMKD"
    },
    {
      "field": "uses",
      "target": "0QPHW703PV"
    },
    {
      "field": "uses",
      "target": "0MXB4CKHFW"
    },
    {
      "field": "uses",
      "target": "0D5YHQGDJY"
    },
    {
      "field": "uses",
      "target": "0A26X79RTB"
    },
    {
      "field": "uses",
      "target": "0SY9ECQ19Q"
    },
    {
      "field": "uses",
      "target": "04Z5G0THN6"
    },
    {
      "field": "uses",
      "target": "01X5CVV1Q5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/542SG6445R.md",
  "msc": [
    "54F65"
  ],
  "name": "Proof that the Cantor set is homeomorphic to the space of binary sequences",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/542/542SG6445R.md",
  "status": null,
  "tag": "542SG6445R",
  "uri": "m2://542SG6445R",
  "url": "https://math2.tamnd.com/wiki/proof-that-the-cantor-set-is-homeomorphic-to-the-space-of-binary-sequences/",
  "writing": {
    "tag": "542SG6445R",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}