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So I've just picked up topology and i read that for any set $X$ the entire set is an element of every topology and so is an open set. but why is this so? I understand that the complement of $X$ is $\emptyset$ which is open by vacuous truth however how do we then show that $X$ is open?

Thanks in advance.

cmk
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The homeschooler
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    How do you define open? How do you define topology? – supinf Jan 21 '21 at 16:25
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    Well you can either define $X$ to be an open subset or to require that the open sets cover $X$. You want every point to have some neighbourhood. Since arbitrary unions of open sets are open, this requirement forces $X$ to be open. – Tyrone Jan 21 '21 at 18:32

3 Answers3

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By definition of a topology the set $X$ and $\emptyset$ will lie in it, so that they are always both open and closed.

Someone
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  • @i don't know i understand that they are in the topology by definition but why what was the idea of including them in every topology – The homeschooler Jan 21 '21 at 16:15
  • If you think about it for a set $F$ to be open in the topology generated by a basis we need that for every $x\in F$ there exists a basis element $U$ such that $x\in U\subset F$. Now this will obviously be true for $F=X,\emptyset$. And so I think it's just natural that $X$ and $\emptyset $ are always open sets even if we consider a topology that is not generated by a basis , and if there are two canonical sets to consider these two are the ones since they always exist no matter how weird the set $X$ is. – Someone Jan 21 '21 at 16:17
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Given a set $X$, a topology is a collection of subsets of $X$ that satisfy some "axioms". Those axioms define what a topology is on $X$. One of those axioms says that both $X$ and the empty set are in the topology. This is how you "show" that $X$ is always open with respect to any topology.

Given any set $X$, it has the trivial topology which consists of only the empty set and the entire set $X$.

Notes.

It is a reasonable question to ask why those axioms, for instance why the whole space is always assumed to be open in the axioms. You may want to read this question: Why the axioms for a topological space are those axioms?

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It is possible to define topology by means of 'neighborhoods' in the first place.

Then a set $A$ is open if every point of $A$ has a neighborhood $B$ fully in $A$: $B\subseteq A$.

Since all these sets, including the neighborhoods (whatever they are) live inside $X$, the ambient set $X$ will satisfy this definition.

However, writing out the axioms for the neighborhood approach and working rigorously based on them would be more complicated than its cleaned up version that puts the open sets in the first place, and write up their basic properties as axioms.

Berci
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  • If "$X$ is open" were deleted from the definition, you could have cases where some points had no neighbourhoods at all. These would correspond, in the actual definition of topology, to points whose only neighbourhood is $X$. There certainly are topological spaces that have such points, but they aren't particularly useful or interesting AFAIK. – Robert Israel Jan 21 '21 at 19:58