Open sets

Definition: Open set in a metric space

A set $E\subset X$ is open if every point of $E$ is an interior point. More precisely, for all points $x\in E$, there exists $r>0$ such that the $r$-neighborhood or $r$-ball about $x$ given by

$$B(x,r)=\{y\in X|d(x,y)<r\},$$

where $d$ is a metric on $E$, is completely contained in $E$.

^956753 We have the following facts about open sets:

• The $r$-ball about a point $x$, as defined above, is an open set.
• If $f:X\to Y$ is a continuous function and $U\subset Y$ is open, then the preimage $f^{-1}(U)$ is open as well.
• $f$ is continuous if and only if for all open sets $U\subset X$, the image $f(U$) is an open set (see also: equivalent definitions of continuity in metric spaces).

Definition: Open set in a topological space

If $X$ is a topological set with topology $\mathcal{T}$, we say a subset $U\subset X$ is an open set of $X$ if $U\in \mathcal{T}$.

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Related: Closed sets and closures, Topological spaces and open sets

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Review §

Exercises §

Modern Analysis I

• Using the definition of an open set in a metric space, prove that arbitrary unions and finite intersections of open sets are open. ⭐

MATH-42X

• Determine whether the following intervals in $\mathbb{R}$ are open, closed, both, or neither:
  • Bounded: $(a,b)$, $[a,b)$ $(a,b]$, $[a,b]$
  • Unbounded: $(a,\infty )$, $[a,\infty )$ $(-\infty ,a)$, $[-\infty ,a]$
  • $\varnothing$ and $\mathbb{R}$
• Given a finite collection of open sets, is their intersection open or closed? Justify.
  • Does this result also hold for infinite (countable) intersections? Give an example or counter-example. ⭐
• Prove that every ball is open. ⭐

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Notes §

• The definition for open set used above is a generalization of the following definition from ${\mathbb{R}}^n$ to metric spaces:

Definition: Open set in \mathbb R^n

A set $U\subset {\mathbb{R}}^n$ is open if for all ${\vec{x}}_0\in U$, there exists $ϵ>0$ such that

$$\Vert \vec{x}-{\vec{x}}_o{\Vert }_2<ϵ⟹\vec{x}\in U.$$

#wip Define interior point