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 $f(\\overline{A})=\\overline{f(A)}$ .
随便看	SyntacticCongruence syntactic congruence SyntacticPropertiesOfANormalModalLogic syntactic properties of a normal modal logic SyntheticDivision synthetic division SyntopogenousStructure syntopogenous structure SystematicSampling systematic sampling SystemModel system model SystemOfDistinctRepresentatives system of distinct representatives SystemOfOrdinaryDifferentialEquations system of ordinary differential equations SystemState system state SzemeredisTheorem SzemerediTrotterTheorem Szemerédi-Trotter theorem Szemerédi’s theorem T0Space T0 space T1Space