properties of the closure operator

Suppose $X$ is a topological space, and let $\\overline{A}$ be theclosure of $A$ in $X$ .Then the following properties hold:

1.
$\\overline{A}=A\\cup A^{\\prime}$ where $A^{\\prime}$ is the derived set of $A$ .
2.
$A\\subseteq\\overline{A}$ , and $A=\\overline{A}$ if and only if $A$ is closed
3.
$\\overline{A}=\\emptyset$ if and only if $A=\\emptyset$ .
4.
If $Y$ is another topological space, then $f\\colon X\\to Y$ is a continuous map,if and only if $f(\\overline{A})\\subseteq\\overline{f(A)}$ for all $A\\subseteq X$ . If $f$ is also a homeomorphism,then $f(\\overline{A})=\\overline{f(A)}$ .

单词	PropertiesOfTheClosureOperator
释义	properties of the closure operator Suppose $X$ is a topological space, and let $\\overline{A}$ be theclosure of $A$ in $X$ .Then the following properties hold: 1. $\\overline{A}=A\\cup A^{\\prime}$ where $A^{\\prime}$ is the derived set of $A$ . 2. $A\\subseteq\\overline{A}$ , and $A=\\overline{A}$ if and only if $A$ is closed 3. $\\overline{A}=\\emptyset$ if and only if $A=\\emptyset$ . 4. If $Y$ is another topological space, then $f\\colon X\\to Y$ is a continuous map,if and only if $f(\\overline{A})\\subseteq\\overline{f(A)}$ for all $A\\subseteq X$ . If $f$ is also a homeomorphism,then $f(\\overline{A})=\\overline{f(A)}$ .
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