Clarification on Sets in Topological Spaces being Open In the definition of a topological space, say $(X,\tau)$, the elements of $\tau$ are called open sets. Of course, for abstract topological spaces, this is a convenient definition as we might not have a concrete definition of openness, at least not in the way that we're able to define openness in metric spaces, but for topological spaces that are, for example, comprised of subsets of $\mathbb{R}$, do the elements of $\tau$ necessarily need to be open in the "usual sense" in order for them to actually be elements of $\tau$, or can they be closed, as well? Intuitively, I think the answer (for $\mathbb{R}$, at least) is no, because of two reasons:

*

*Let $\tau=\lbrace[a,b]\subset\mathbb{R}:(a,b)\in\mathbb{R}^2,~a<b\rbrace$, then we have that
$$
\bigcup_{k=1}^\infty[1/k,2-1/k]=(0,2)\notin\tau
$$
so $\tau$ cannot, in this case be a topological space.

*Let $\tau=\lbrace(a,b)\subset\mathbb{R}:(a,b)\in\mathbb{R}^2,~a<b\rbrace$ as follows
$$
\bigcup_{k=1}^\infty(-1/k,2+1/k)=[0,2]
$$
So ultimately, even though there is a way to construct candidate topological spaces comprised either entirely out of open sets, or entirely out of closed sets, neither of these are actually topological spaces, which leads me to believe that topological spaces in $\mathbb{R}$ must be somewhere in-between, meaning that even thought subsets of $\mathbb{R}$ may be closed in the "usual sense", they can still be regarded as open sets, provided they form part of a topological space. Is there any sense to this reasoning? Any help with this concern is appreciated.
 A: The easiest example is the discrete topology (generated by all singletons). Each singleton is closed in the usual sense (generated by open intervals), but is clopen (open and close) in any discrete topology. In discrete topology, every subset is clopen.
A: Your instincts are correct. The answer is no. In general, open sets in an arbitrary topology do not correspond to open sets in the standard topology.
In fact, the definition of a topology as a collection of open sets is an abstraction of the collection of concrete open sets in $\mathbb R^n$.
The open sets in $\mathbb R^n$ have the following properties.

*

*The empty set and the entire set are both open.

*Arbitrary unions of open sets are open.

*Finite intersections of open sets are open.

Notice that these properties are what we use to define a topology.
It is a common practice in mathematics to take a set of properties of a concrete object and define other objects as things that share these properties. That is what an abstract topology is. This abstraction is useful because we can show that a continuous function, defined in terms of $\epsilon-\delta$, is equivalent to a definition in terms of open sets.
