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Can a set be closed in one topology but neither open nor closed in another?

If we say that the complement of a open set is a closed set, i.e. if $S \subseteq X$ is open then $X \setminus S$ is closed, I argue that depending on the topology, it could actually be the case. Consider the Euclidean topology on $\mathbb{R}$. Clearly any closed interval $[a,b]$ with $a,b \in \mathbb{R}$ is a closed set in this topology.

However, given the trivial topology on $\mathbb{R}$, any interval $[a,b]$ is neither closed or open.

But according to this Wikipedia page, an equivalent definition of a closed set is that "a set is closed if and only if it contains all of its limit points". And from this point of view, the inverval $[a,b]$ is clearly closed.

So i guess that my question is: How can two (supposedly) equivalent definitios of a closed sets, have a set be closed in one and not the other?

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    $\begingroup$ Try the discrete topology as starting topology. $\endgroup$ Commented Mar 3, 2015 at 20:36

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The definition of a limit point depends on the topology too.

Recall the definition of a limit point:

A limit point of a set $S$ in a topological space $X$ is a point $x$ (which is in $X$, but not necessarily in $S$) that can be "approximated" by points of $S$ in the sense that every neighbourhood of $x$ with respect to the topology on $X$ also contains a point of $S$ other than $x$ itself.

With this definition, and the trivial topology on $\mathbb{R}$, any point of $\mathbb{R}$ is a limit point of $[a, b]$.

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  • $\begingroup$ Thank you. I think the critical point that i missed is that the definition of limit points depends on the topology too. $\endgroup$
    – AcId
    Commented Mar 3, 2015 at 12:17
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Take $[0,1]$. It is closed in $\mathbb{R}$ equipped with the usual topology. Now equip $\mathbb{R}$ with the trivial topology $\{\emptyset, \mathbb{R} \}$. The complement of $[0,1]$ is not $\emptyset$ neither $\mathbb{R}$, so it is not open, therefore $[0,1]$ is not closed in this topology.

Choosing a topology is precisely choosing what sets are open and (equivalently) what sets are closed.

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in discrete topology any set is closed as well as open... so $\mathbb{R}$ with discrete topology $[0,1)$ is open as well as closed but with resprct to euclidean topology $[0,1)$ nither closed nor open

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A topology is a declaration of which sets are closed, and in general, whether a set is closed in one topology has nothing to do with whether it is closed in another topology.

For example, consider any set $X$ with at least $2$ elements. Then, any set $U$ such that $\emptyset \subsetneq U \subsetneq X$ is closed in the discrete topology (in which all subsets of $X$ are closed) but not in the trivial topology (in which only $\emptyset$ and $X$ itself are closed).

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