irreducible

A subset $F$ of a topological space $X$ is reducible if it can be written as a union $F=F_{1}\\cup F_{2}$ of two closed proper subsets $F_{1}$ , $F_{2}$ of $F$ (closed in the subspace topology). That is, $F$ is reducible if it can be written as a union $F=(G_{1}\\cap F)\\cup(G_{2}\\cap F)$ where $G_{1}$ , $G_{2}$ are closed subsets of $X$ , neither of which contains $F$ .

A subset of a topological space is irreducible (or hyperconnected) if it is not reducible.

As an example, consider $\\{(x,y)\\in\\mathbb{R}^{2}:xy=0\\}$ with the subspace topology from $\\mathbb{R}^{2}$ . This space is a union of two lines $\\{(x,y)\\in\\mathbb{R}^{2}:x=0\\}$ and $\\{(x,y)\\in\\mathbb{R}^{2}:y=0\\}$ , which are proper closed subsets. So this space is reducible, and thus not irreducible.

单词	Irreducible1
释义	irreducible A subset $F$ of a topological space $X$ is reducible if it can be written as a union $F=F_{1}\\cup F_{2}$ of two closed proper subsets $F_{1}$ , $F_{2}$ of $F$ (closed in the subspace topology). That is, $F$ is reducible if it can be written as a union $F=(G_{1}\\cap F)\\cup(G_{2}\\cap F)$ where $G_{1}$ , $G_{2}$ are closed subsets of $X$ , neither of which contains $F$ . A subset of a topological space is irreducible (or hyperconnected) if it is not reducible. As an example, consider $\\{(x,y)\\in\\mathbb{R}^{2}:xy=0\\}$ with the subspace topology from $\\mathbb{R}^{2}$ . This space is a union of two lines $\\{(x,y)\\in\\mathbb{R}^{2}:x=0\\}$ and $\\{(x,y)\\in\\mathbb{R}^{2}:y=0\\}$ , which are proper closed subsets. So this space is reducible, and thus not irreducible.
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