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Symmetric product (topology)
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<p>In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the <i>n</i>th symmetric product of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> consists of the unordered <i>n</i>-tuples of its elements. If one fixes a <a href="/facts/Pointed_space/V4DTgRCd">basepoint</a>, there is a canonical way of <a href="/facts/Embedding/HKeVKc42">embedding</a> the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the <a href="/facts/Colimit/JoLugLs3">colimit</a> over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor. </p><p>From an algebraic point of view, the infinite symmetric product is the <a href="/facts/Free_commutative_monoid/h3nPVjqr">free commutative monoid</a> generated by the space minus the basepoint, the basepoint yielding the identity element. That way, one can view it as the abelian version of the <a href="/facts/James_reduced_product/tBK6ovuo">James reduced product</a>. </p><p>One of its essential applications is the <a href="/facts/Dold-Thom_theorem/pSe6z5Hf">Dold-Thom theorem</a>, stating that the <a href="/facts/Homotopy_groups/xjogL95Y">homotopy groups</a> of the infinite symmetric product of a connected <a href="/facts/CW_complex/rN7buLMo">CW complex</a> are the same as the <a href="/facts/Reduced_homology/ySoPPB63">reduced homology</a> groups of that complex. That way, one can give a homotopical definition of <a href="/facts/Homology_(mathematics)/VFJmryEW">homology</a>. </p>