Q: A question on the relation between the "topology" and the "set" Given a set $X$ and a topology $\tau$ on it. Why are we called the points of $X$ with "elements" of $X$, and why $\tau$ the "open set" of $X$ and not just a topology? The same question was asked here: The question is not quite the same as mine but may be connected. What is the meaning of being an element of a set? But I cannot see the connection of the two. Maybe it is connected to the definition of a topology of a set, which is: A collection of subsets of $X$ which is closed under finite intersections and arbitrary unions. So if there is a set-theoretical answer to the above question, I need it. I want to know why $\tau$ is the "set" of the elements of $X$ and not just a collection of subsets of $X$? A: The difference is rather vague at some levels, but you are trying to get your hands on something more specific than a large categorical object with its own specific set of objects. Now there is a specific set $X$, then, the set $X$ has certain properties, and $X$ (a set) may be a function of the set $X$ (a set). So far so good. Then, from this set $X$ we may want to construct something that "encompasses" $X$. This is not that straight forward to do in a meaningful way. In a way, it is still just as simple to state that we are doing, but it is just a different way of saying it. The topology is not much more than a certain kind of open set. It is some set that includes all subsets of itself. We call these closed under arbitrary unions. This means that they are large enough to include any open set. In this case $X$. This may seem to be a bit vague, but as we start dealing with more complicated sets, and more complicated open sets, we will be in good shape to see this notion of openness again. The notion of an open set that is "arbitrarily large enough to include any open set" is just something like a "basis" for a collection of open sets. It is a small collection of open sets that is large enough to contain all other open sets. It is called "open", and we call it "open", because the union of all these sets is in fact the set $X$ itself. It is as large as possible. And it is a useful set. A function from a set $X$ to $Y$, just means that we have a function for a subset of $X$, not $X$ itself. A: Let $X$ be a set. Let $\tau$ be a collection of subsets of $X$. Then $\tau$ is a topology on $X$ if it satisfies the following conditions: the empty set and the whole set $X$ belong to $\tau$, any union of sets in $\tau$ is also in $\tau$, any finite intersection of sets in $\tau$ is also in $\tau$. Let's check which of these properties actually corresponds to each part of your question. In the definition of topology, what's being referred to is sets, not elements. That is, what is being referred to is what we're calling the set $X$ as a whole, rather than an element of $X$, which is more like what we call $x \in X$ rather than $x \in \tau$. The word "topology" refers to a collection of open sets. This is what we are calling a "topological space". So if we wanted to, we could change the word "topology" in your question, or in the question you linked to, to "topology" rather than "open set", to make more clear that we are talking about sets and not elements, but doing so would seem pretty unnecessary. If there are elements of $X$, then they are "elements of $X$" in some specific sense. But it would probably be unwise to try to answer this question for more specific information about what this sense might be. A: A topological space $X$ can be seen as a set which is endowed with a collection of subsets of $X$ that satisfy some axioms. This collection of subsets is denoted by $\tau$ or $\mathcal{T}$. When we consider the particular case where $X$ is a set of (abstract) points $\mathcal{A}$, the collection $\tau$ is said to be a topology on $X$. Another way to explain it is to say that each $a \in X$ is equipped with a certain topology: a set $\tau_a$ of subsets of $X$ such that if $b \in \tau_a$ then $a \in \tau_b$. This is true for all $a \in X$ and $\tau_a \neq \varnothing$. Now one might ask what's the relationship between the topology $\tau$ and the points $x\in X$. But this leads us to the answer that a particular $\tau$ in general does not determine which point has which topology, although one might say that each point will be equipped with a certain topology (for example, the discrete topology). The following is the definition of topological space (from Wikipedia): Let $X$ be a set. A topology on $X$ is a collection $\tau$ of subsets of $X$, which satisfies the following conditions: the empty set and the whole set $X$ belong to $\tau$, any union of sets in $\tau$ is also in $\tau$, any finite intersection of sets in $\tau$ is also in $\tau$. It turns out that this definition also allows us to answer your question: each point $a \in X$ has a certain topology $\tau_a$ on itself. In fact, in any topological space $X$, the collection $\mathcal{T}$ of open sets satisfies the following: $\varnothing, X \in \mathcal{T}$ if $U \cup V$ is open, then $U$ and $V$ are also open. if $U \cap V$ is open, then $U$ and $V$ are also open. In other words, any $U \in \mathcal{T}$ can be written as the union of smaller subsets of $X$. This definition is satisfied for the empty set $\varnothing \in \mathcal{T}$. So $\varnothing$ can be considered to be the union of its elements, which is empty itself, which is clear from the empty set definition. But in order to allow the discrete topology we need to assume that the empty set is open, which leads to the first axiom of topology.