4:["$","div",null,{"className":"min-h-screen bg-white flex flex-col","children":[["$","div",null,{"className":"px-6 pt-6 pb-4 border-b border-gray-100","children":[["$","div",null,{"className":"flex items-center gap-1.5 text-xs text-gray-400 mb-4","children":[["$","a",null,{"href":"/","className":"hover:text-[#26a69a] transition-colors","children":"Market"}],["$","span",null,{"children":"›"}],["$","span",null,{"className":"text-gray-600","children":"Tychonoff plank"}]]}],["$","div",null,{"className":"text-[10px] text-gray-400 uppercase tracking-widest mb-1","children":"Company Profile"}],["$","h1",null,{"className":"text-2xl font-bold text-gray-900","children":"Tychonoff plank"}],null]}],["$","div",null,{"className":"flex-1 px-6 py-6","children":[["$","p",null,{"className":"text-sm text-gray-600 leading-relaxed mb-8","children":"In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces and , where is the first infinite ordinal and the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point ."}],["$","div",null,{"className":"columns-1 md:columns-2 gap-10","children":[["$","div","0",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Definition"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"Let \\Omega be the set of ordinals which are less than or equal to \\omega and \\Omega_1 the set of ordinals less than or equal to \\omega_1 . The Tychonoff plank is defined as the set \\Omega \\times \\Omega_1 with the product topology. The deleted Tychonoff plank is the subset S = \\Omega \\times \\Omega_1 \\setminus \\{ (\\omega,\\omega_1) \\} , where S is the plank with a corner removed. ==Properties=="}}]]}],["$","div","1",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Properties"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal. The Tychonoff plank is not perfectly normal because it is not a Gδ space: the singleton \\{\\infty\\} is closed but not a Gδ set. The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank. == See also =="}}]]}]]}],["$","div",null,{"className":"mt-8 pt-4 border-t border-gray-100","children":["$","a",null,{"href":"https://en.wikipedia.org/wiki/Tychonoff%20plank","target":"_blank","rel":"noopener noreferrer","className":"text-xs text-gray-400 hover:text-gray-600 transition-colors","children":"Source: Wikipedia ↗"}]}]]}],["$","div",null,{"className":"px-6 py-4 border-t border-gray-100 flex justify-between items-center","children":["$Lf","$L10"]}]]}]