What is an open set in a topological space? I recently started learning topology to help me understand limits and continuity better for calculus, and I am struggling with some of the definitions.
What I am getting confused with is why is every set in a topology considered to be open and when talking about sets in the topology we always say the set is open.
My intuitive notion of openness from  previous knowledge of mathematics is an interval that does not contain its endpoints, so there is an infinite sequence at the end points, e.g., $(0,1)$ is an open interval.
However, in topology, for example, the singleton $\{1\}$
is considered an open set—how is this so? Why are sets in a topology always open? And what is the definition of an open set in a topological space?
Thanks in advance.
There is no such thing as an &quot;open set,&quot; only a open subset of a topological space. &quot;The singleton $\{ 1 \}$ is an open set&quot; is not a meaningful statement; the meaningful statement is whether the singleton $\{ 1 \}$ is an open subset of some topological space containing $1$. For example it is not an open subset of $\mathbb{R}$, or of the closed interval $[0, 1]$ (with their usual topologies). However, it is an open subset of itself, considered as a one-point space (which has a unique topology).
Sets in a topology are open by definition; that's what &quot;open&quot; means.
I suggest that when starting to learn topology you completely ignore the ordinary English meaning of the word &quot;open&quot; and just work with the axioms abstractly for awhile. It's not the most fun or intuitive way to do it but at least you aren't letting preconceptions get in the way. Also, I don't really recommend this as a way to better understand calculus; you'd be better off picking up a textbook on real analysis.
There are a number of things about topology that are idiosyncratic. It is typical to refer to singleton sets with the same symbol as used for the element of the singleton. This should act as an indication that one is actually working within the power set of the domain of definition.
A topology is given as a partition on the power set. Provided that a partition is compatible with the axioms, the partition will consist of open, closed, clopen (both open and closed) and none of the above. The axioms can be given in terms of either open sets or closed sets. So, these classifications reside in the power set of the power set of the domain of definition. Since one can have clopen sets, this classification does not produce disjoint collections.
As a set, the domain of definition has a sort of special relationship with the topology in which every singleton is open. This is called the discrete topology. Every set is open by arbitrary unions. So every set is closed as well by complementation.
Importantly, one can have topologies where not every set is open. The topology with only the empty set and the domain of definition as open sets satisfies the axioms. It is called the trivial topology.
From these remarks, you should surmise that &quot;what is open&quot; in a topology is a matter of stipulation.  For real analysis and calculus the stipulation comes from our naive conceptions of length and measurement. What becomes important are inequalities. The &quot;in between&quot; mathematical object is called a metric space. This governs how we understand distances between points.
Whatever topology book you are using, it could be highly useful to obtain a copy of &quot;Counterexamples in Topology&quot; by Steen and Seebach (reprints from Dover Books are inexpensive).  It is full of examples, including some simple ones accessible to beginners. It also summarizes what you will learn from main textbook. So, it will become a reference resource as you learn the subject better.