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.
 A: There is no such thing as an "open set," only a open subset of a topological space. "The singleton $\{ 1 \}$ is an open set" 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 "open" means.
I suggest that when starting to learn topology you completely ignore the ordinary English meaning of the word "open" 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.
A: The whole point of having a general topology is that you get to define which sets are and aren't "open", to make the rules of the game, and then get to see what that does and how things are different in the "world" so created versus the usual real numbers. If we take the idea that an open set "doesn't contain its own boundary", which is what you are after but how I originally heard it phrased, being able to define open sets to be whatever you want them to be (so long as you meet the rules for how they must be structured under union and intersection) means, in effect, you get to define what constitutes a "boundary" and what doesn't. You get to make what is and isn't an "end point".
To see why that has an impact, note that the only reason that $0$ and $1$ are "boundaries" of $(0, 1)$ is because of the ordering on the reals, which ensures that $0 < x < 1$ whenever $x \in (0, 1)$, and also, there's nothing in between 0 and 1 and the set $(0, 1)$, i.e. no points $y$ such that $0 < y < x$ for every $x \in (0, 1)$, and similarly for $1$.
But suppose we re-ordered the reals, so that both points $0$ and $1$ came before the points we consider to be in $(0, 1)$ (in the usual definition.). E.g. suppose we ordered the reals to look like
$$(\text{stuff}) < 0 < 1 < 2 < (\text{stuff}) < (\text{the numbers in $(0, 1)$}) < (\text{more stuff})$$
Now, suddenly, $(0, 1)$ no longer has boundary points $0$ and $1$. So there is no absolute notion of a "boundary point". It depends on the order, and we just redefined what the boundary was by redefining the order.
And topology is even more flexible than that. And orders are just one source, but far from the only one, of topologies.
A: Consider the set $\mathbb{R}$. A topology on $\mathbb{R}$ is a collection* of subsets on $\mathbb{R}$ that we declare to be open.
For example, the usual topology on $\mathbb{R}$ consists of all open intervals like $(0,1)$ and $(a,b)$ and all unions of such intervals. In this topology, the singleton set $\{1\}$ is not open, since it cannot be written as a union of open intervals $(a,b)$.
There are other topologies on $\mathbb{R}$. We could, for example, declare that every subset of $\mathbb{R}$ is open. In other words, the topology consists of all subsets of $\mathbb{R}$. In this topology, $\{1\}$ is open.
As another example, we could declare that the only open sets are $\mathbb{R}$ and $\emptyset$. For this topology, $\{1\}$ is not open. Also, $(0,1)$ is not open.
There are many other topologies that we can put on $\mathbb{R}$. Of course, some are more useful and interesting than others.
*Note: We can't just grab an arbitrary collection of subsets $\mathbb{R}$ and call it a topology. The collection needs to contain $\mathbb{R}$ and the empty set and it needs to be closed under arbitrary unions and finite intersections.
There is nothing special about $\mathbb{R}$ here. Given an arbitrary set $X$, a topology on $X$ is a collection of subsets of $X$ (that contains $X$ and $\emptyset$ and is closed under arbitrary unions and finite intersections). The sets in the topology are called open sets of that topology.
A: An open set is an abstraction.  So an "open" set, for the most part, whatever we want it to be.  And a Topology is nothing more or less than a list of sets we say "These ones; these are the sets that we are going to be considered open.
We can make the list really really short by having nothing in the list (except the empty set  and the universal set); or we can make the list really really long by having every possible set in the list.  Or we could make it completely arbitrary by saying sets with only elements wearing purple socks will be "open".
The only rules are:  1) The empty set and the universal set must both be open. 2) Any finite intersection of open sets is open; 3) any union (finite or infinite) of open sets is open.
So if for instance our universal set is $\{Heidi, Mary, Sam, Bill\}$ and our topology is $\{\{Mary,Bill\}, \{Heidi, Sam\}, \{Heidi, Bill\}\}$ then you we have $16$ sets and of them:
$\emptyset$ is open because that's the law.
$\{Heidi\}$ is open because it is the intersection of $\{Heidi,Sam\}$ and $\{Heidi,Bill\}$
$\{Mary\}$ is not open because it's not the union or intersection of any open sets.
$\{Sam\}$ is not open because it's not the union or intersection either.
$\{Bill\}$ is open because it's the intersection of $\{Mary,Bill\}$ and $ \{Heidi, Bill\}$
$\{Heidi, Mary\}$ is not open because we just can't get it from the rules.
$\{Heidi, Sam\}$ is open because we said it was.
$\{Heidi, Bill\}$ is open because we said it was.
$\{Mary, Sam\}$ is not open because we can't get it from the rules.
$\{Mary, Bill\}$ is open because we said it was.
$\{Sam,Bill\}$ is not open because we can't get there.
$\{Heidi, Mary, Sam\}$ is not open.
$\{Heidi, Mary, Bill\}$ is open because it is the union of $\{Heidi, Mary\}$ and $\{Heidi, Bill\}$
$\{Heidi, Sam, Bill\}$ is open because it is the union of $\{Heidi, Sam\}$ and $\{Heidi, Bill\}$.
$\{Mary, Sam, Bill\}$ is not  open.
And $\{Heidi, Mary, Sam, Bill\}$ is open because it is the unversal set and it's the law that it must be open.
....
That's it.  Open means what we say it does.
.....
The real question is if in a metric space to claim "open" means "every point is an interior point" we have to ask: Does that follow the rules?  That is:
Are the empty set and the universal set open?  (Empty set is vacuously open because it has no points so all points are interior points; universal set is open as every open ball is a subset of the universal set so every point is an interior point of the universal set.)
Is the finite intersection of any sets where every point is an interior point itself has that property?  (It does.... It requires a proof though)
Is every union of sets with that property have that property? (ditto).
And finally, do no other set have that property?.... (yeah, it's tedious be it can be proven).
.....

I recently started learning topology to help me understand limits and continuity better for calculus

If it helps you, you will probably be the first.
Limits and real numbers and distances in a metric space make for a very very intuitive topology.  But An abstract topology will probably be hard to "see" off hand.  (Does my example of four people and some groups are open and others are not make any sense?  No, it does not... but it follows the rules!)  What one hopes to get from this is how to verify solutions based entirely on the logic of rules and not instinct or (much worse) assumptions (which can often be dead wrong.)
A: I have read above interesting answers! Here I would like to share a way how I understand the generalized notion open set from the familiar notion open interval.
Imagine that you are a mathematician! LET'S GO!!!
Part 1. What is "Topology"?
Now, you want is to give a definition for "open set" (in a general set, for example, a discrete set), which generalizes the open intervals $(a,b)$ (in real line). What will you do?
...thinking...
Yes, exactly! You have to collect the key properties of open intervals!
To obtain the key properties, the most important thing is to distinguish the open interval $(a,b)$ and the closed interval $[a,b]$. (Actually, we should compare it with half-open intervals $[a,b)$ and $(a,b]$, but we shall ignore them to avoid the unnecessary complexity ^o^).
The first  difference came to our mind is probably the appearance of the "end points" $a, b$ in $[a,b]$ while $(a,b)$ doesn't have. Now, how to generalize the notion of "end points" to a discrete setting? It seems to be challenging!
...Let's think in a different way...Let's think creatively...
Eureka! Ohlala!

Let $\mathcal T$ be the set of all open intervals (loosely speaking!)

It means: a set is open if and only if it belongs to $\mathcal T$! Now, we only need to study the properties of $\mathcal T$, which is a collection of sets. This approach seems to be promising to defined an open in discrete setting as we desired! That's so cool! Right? Hahaha!
Let's give $\mathcal T$ a name!

$\mathcal T$ is called a topology.

Yup, what a super ultra cool name!
Part 2. The characteristics of Topology.
When we study a collection of sets. What we should do is to investigate the intersection and the union . Let's start!
Part 2.1. The intersection.
Now, look at the intersection of two sets
\begin{align}
(a,b)\cap (c,d)& =(b,c)\\
[a,b]\cap [c,d]& =[b,c]
\end{align}
for $a<c<b<d$. It means the intersection of two open intervals (res. closed intervals) is also an open interval (res. closed interval). So what is the difference here? Nothing! Let's consider  an arbitrary intersection
\begin{align}
\bigcap_{n=1}^\infty (0,1+\frac{1}{n})=(0,1]\\
\bigcap_{n=1}^\infty [0,1+\frac{1}{n}]=[0,1].
\end{align}
Oh la la, this is exactly the difference that we are looking for! It means that the arbitrary intersection of open intervals is not an open interval.
Hence, there is a difference between finite intersection and arbitrary intersection! Keep this in mind!
Now, we may consider this fact as a feature of $\mathcal T$.

Intersection of two (or finite) open intervals is again an open interval.

Part 2.2. Union.
Analogously, we find that
\begin{align}
\bigcup_{n=1}^\infty (0,1-\frac{1}{n})=(0,1)\\
\bigcup_{n=1}^\infty [0,1-\frac{1}{n}]=[0,1).
\end{align}
and claim another feature of $\mathcal T$ is

Union of arbitrary open intervals is still an open interval.

Part 2.3. Trivial sets.
Now, consider the empty set and the whole real line.
Since they cannot be written as $(a,b)$ nor $[a,b]$, how can we give them a reasonable property, open or closed? From the intersection part, we expect that intersection of any two open sets is again an open set, and clearly, the intersection may be empty. Thus, it is reasonable to define the empty set as an open set! Similarly, from the union part, it is reasonable to define the whole set as an open set! Overall, we may state that

The empty set and the whole set are open.

Part 3. Every set can be open in some sense.
Let $X$ be a set, it may be a real line or any discrete set.
Now, any subsets of $X$ can be considered as an open set if it belongs to a collection satisfying the three properties above, and we call such collection a "topology"! Conversely, "topology" is just a collection of open sets.
In a nutshell, any set (including the singleton) will be open as soon as you can put it into a collection, named topology, satisfying the three properties.
A: 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 "what is open" 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 "in between" 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 "Counterexamples in Topology" 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.
