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Definition: Topological Sum, Disjoint Union

Let $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ be two topological spaces. The disjoint union of $X$ and $Y$ is defined by $$X+Y:=X\times \{1\}\cup Y\times \{2\}.$$

The topological sum $(Z,\mathcal O_Z)$ of $X$ and $Y$ is defined by $$(Z,\mathcal O_Z):=(X+Y,\mathcal O_X\cup\mathcal O_Y).$$

Notes

• The disjoint union is a formal trick ensuring that the set union $\cup$ does not get rid of duplicate common elements of $X$ and $Y$. In particular, $X+X$ is not the same as $X\cup X=X.$

Mentioned in:

Propositions: 1 2

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References

Bibliography

1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition