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      "title": "Tychonoff's Theorem at a glance",
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      "transcript": "TOPOLOGY · ARBITRARY PRODUCTS\nTYCHONOFF'S THEOREM\n∀ i, IsCompact(s(i))\n⇒ IsCompact {x | ∀ i, x(i) ∈ s(i)}\nARBITRARY INDEX TYPE ι\nSAMPLE LABELS i₁, i₂, i₃, iα\nX(i₁), X(i₂), X(i₃), X(iα)\ns(i₁), s(i₂), s(i₃), s(iα)\nONE TOPOLOGICAL SPACE X(i) PER INDEX\nONE COMPACT SUBSET s(i) ⊆ X(i)\nDEPENDENT FUNCTIONS x\nEVERY COORDINATE SATISFIES x(i) ∈ s(i)\nTHE CONSTRAINED PRODUCT SUBSET IS COMPACT\nEXACT SCOPE\nCompactness of the subset. No nonemptiness claim."
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      "caption": "Compact coordinate subsets combine into a compact dependent product; the receding family suggests an unrestricted index set.",
      "description": "On a forest-green engraved field, many coordinate spaces continue toward the horizon, each containing an enclosed glass-like compact region. Fine colored filaments run from those selected regions into a gold lattice held beneath a closed dome, visually separating compact coordinate subsets from the resulting constrained product subset.",
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      "title": "Infinite product compactness schematic",
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          "alt": "An open-ended receding family of engraved forest-green coordinate chambers is threaded by one cobalt braided current inside a fine charcoal product lattice, with empty evaluation apertures but no selected points.",
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          "caption": "Mathlib rewrites compactness, product neighborhoods, and product membership so the proof begins with an ultrafilter containing the constrained product subset.",
          "description": "`isCompact_iff_ultrafilter_le_nhds`, `nhds_pi`, `le_pi`, and `le_principal_iff` transform the compactness goal into a convergence problem for an arbitrary ultrafilter lying over the dependent product subset. This reformulation introduces no coordinate witnesses yet.",
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          "title": "Product compactness becomes an ultrafilter convergence problem",
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          "equation": "f <= principal {x | forall i, x i in s i} implies a required product limit",
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          "alt": "The cobalt braid projects through identical apertures into the receding green chambers, where exactly one antique-gold point receives each visible coordinate strand.",
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          "caption": "Mapping the ultrafilter through each evaluation map preserves membership in the corresponding compact set, so coordinate compactness provides a limit point there.",
          "description": "For each index `i`, Mathlib maps the ultrafilter through `Function.eval i`. Membership of the constrained product implies that the mapped ultrafilter contains `s i`; compactness of `s i` then supplies a point `x i` in that set and convergence of the projected filter to it.",
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          "title": "Every coordinate projection acquires a compact limit",
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          "equation": "exists x, x in s i and Tendsto (Function.eval i) f (nhds x)",
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          "caption": "Dependent choice selects the coordinate limits; their membership and coordinatewise convergence assemble the required point and convergence in the product topology.",
          "description": "`choose x hx using this` forms the dependent function from the coordinate witnesses. The membership component of `hx i` places the resulting function in the constrained product, while the convergence component for every coordinate reassembles through `le_pi` into convergence to that function in the product topology.",
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          "title": "Coordinate limits reassemble into a product limit",
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      "title": "How ultrafilter limits prove product compactness"
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    "whyLandmark": "Tychonoff's theorem is one of topology's central compactness principles. Its arbitrary-product scope is a decisive reason product topology is defined as it is."
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    "The local evidence checks this exact upstream declaration at one pinned Mathlib commit.",
    "The page does not independently classify the theorem's foundational choice principles beyond the recorded axiom report."
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    "locallyReproduced": "Exact upstream declaration replayed",
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  "summary": "An arbitrary product of compact subsets is compact in the product topology.",
  "targetId": "target.library.mathlib.tychonoff-theorem.v001",
  "title": "Tychonoff's Theorem",
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