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  "body": "\n## Definition\n\nLet $(X_i, x_i)_{i \\in I}$ be a family of [[pointed spaces|T9DG8DDE75]].\nThe *wedge sum* of the family is the space\n$$\\bigvee_{i \\in I} X_i = \\Bigl( \\bigsqcup_{i \\in I} X_i \\Bigr) / \\set{x_i : i \\in I}.$$\nIt is the quotient of the [[disjoint union|ZXJKSB30WF]] that collapses the set of base points to one point.\nIt carries the [[quotient topology|0NQ1H2X495]].\nThe image of the base points is the base point of the wedge sum.\n\n## Remarks\n\nFor two spaces one writes $X \\vee Y$.\nEach $X_i$ embeds in the wedge sum, and a pointed map out of $\\bigvee_i X_i$ is the same as a family of pointed maps out of the $X_i$.\nExample: $S^1 \\vee S^1$ is homeomorphic to the figure eight $\\set{z \\in \\C : \\abs{z - 1} = 1 \\text{ or } \\abs{z + 1} = 1}$, where the two circles meet only at $0$.\nNon-example: the disjoint union $S^1 \\sqcup S^1$ without collapsing the base points is not a wedge sum; it is disconnected, while the wedge of connected spaces is connected.\nThe wedge sum depends on the choice of base points in general, but not up to homotopy for well-behaved spaces such as connected CW complexes.\n",
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