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properties of the closure operator (Theorem)

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

  1. $ \overline{A}=A\cup A'$ where $ A'$ 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)}$.
  5. If $ E\subseteq X$ is any set, then
    $\displaystyle A\cap \overline{E} \subseteq \overline{A\cap E}. $



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Cross-references: homeomorphism, continuous map, closed, derived set, properties, closure, topological space
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This is version 7 of properties of the closure operator, born on 2005-05-18, modified 2006-10-16.
Object id is 7075, canonical name is PropertiesOfTheClosureOperator.
Accessed 1612 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
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