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Let $\operatorname{cl}$ is the closure operator for some topology.

I will call induced proximity the proximity defined by the formula:

$$A\delta B\Leftrightarrow \operatorname{cl}(A)\cap\operatorname{cl}(B)\ne\varnothing.$$

Is induced proximity really a proximity for every given topological space?

Also: What I call here induced proximity is the weakest proximity generating our topology, right?

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2 Answers

up vote 1 down vote accepted

From Wikipedia, I learn about proximities:

The resulting topology is always completely regular.

Thus either induced proximity fails to be a proximity if the given topological space is not regular. Or the generated topology may differ from the given topology.

The latter kind of failure occurs in the space $\{1,2\}$ with open sets $\emptyset$, $\{1\}$, $\{1,2\}$. Here, $\{2\}$ is closed in the given topology, but in the generated topology, the closure of $\{2\}$ is $\{x\mid \{x\}\delta\{2\}\}=\{1,2\}$ (because $\operatorname{cl}(\{1\})=\{1,2\}$).

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$\newcommand{\cl}{\operatorname{cl}}$Check the axioms from this answer. It’s immediate that $P_0$ and $P_2$ are satisfied. Suppose that $A\delta(B\cup C)$; then $\varnothing\ne\cl A\cap\cl(B\cup C)=\cl A\cap\big(\cl B\cup\cl C\big)=(\cl A\cap\cl B)\cup(\cl A\cap\cl C)$, so at least one of $\cl A\cap\cl B$ and $\cl A\cap\cl C$ is non-empty, and $A\delta B$ or $A\delta C$; thus, $P_1$ is satisfied.

$P_3$, however, becomes $\cl\{x\}\cap\cl\{y\}=\varnothing$ iff $x\ne y$ in this setting; for this you want $X$ to be $T_1$.

$P_4$ is also problematic. If $A\bar\delta B$, then $\cl A\cap\cl B=\varnothing$. We want to find $C,D\subseteq X$ such that $A\bar\delta C$, $B\bar\delta D$, and $C\cup D=X$, i.e., such that $\cl A\cap\cl C=\cl B\cap\cl D=\varnothing$ and $C\cup D=X$. Setting $U=X\setminus\cl C$ and $V=X\setminus\cl D$, we see that this requires finding open sets $U,V\subseteq X$ such that $\cl A\subseteq U$, $\cl B\subseteq V$, and $U\cap V=\varnothing$, so for this we want $X$ to be normal.

You do get a proximity if $X$ is $T_4$.

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According to Wikipedia, $P_3$ is $A \cap B \neq \emptyset \implies A\delta B$. Since $A \cap B \subseteq \text{cl}(A \cap B) \subseteq \text{cl}(A) \cap \text{cl}(B)$ it follows that if $A \cap B \neq \emptyset$ then $\text{cl}(A) \cap \text{cl}(B) \neq \emptyset$ and so $A\delta B$. Why is $T_1$ necessary? –  Fly by Night Oct 22 '12 at 16:40
@FlybyNight: There is a reason that I linked to this answer: I’m using the same axioms. With a slight renumbering they are the axioms given in Engelking’s General Topology and with yet another numbering, the axioms to be found in Willard’s General Topology. –  Brian M. Scott Oct 22 '12 at 16:46
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