[Math] How to prove that a set is closed using only elements within the set? (Are there any properties specific to members of closed sets?)

general-topologyreal-analysis

To demonstrate that a set is open, you can show that any element of it is an interior point. That is, for any element $x$ of an open set $S$ there exists some ball of center $x$ and positive radius. Likewise, if you find that all elements of a set are interior points, you can figure that the set is open.

What property can be used to show that a set is closed? By a definition of closed sets you could show that its complement is open, but that method involves elements outside of the set.

Are there any properties specific to elements of closed sets?

Best Answer

You can’t limit yourself entirely to points of the closed set, because whether a set is closed depends on how it interacts with the surrounding space. For example, $\Bbb Q\times\{0\}$ is a closed subset of $\Bbb Q^2$ but not of $\Bbb R^2$. You don’t, however, have to look just at the complement.

Let $X$ be a topological space and $F\subseteq X$. Then the following are equivalent:

  • $F$ is closed in $X$.
  • If $\nu$ is a net in $F$ that converges to some $x\in X$, then $x\in F$.
  • If $\mathscr{F}$ is a filter on $F$ that converges to $x\in X$, then $x\in F$.

Note that both the net and the filter versions deal specifically with points of $F$, though they (necessarily) also refer to points that may be outside $F$, namely, the limit points of the nets or filters.

This PDF is a good introduction to nets and filters in topology.

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