# Cocountable Topology

Given a space $X$, there are many ways to define a topology. More common methods are one of the following:

• explicitly list out all the open sets
• define the rules for a subset $U \subseteq X$ to be open
• define the basis/subbasis of the topology (Note: The basis here refers to topological basis, this is different than a vector space basis that we have learnt in linear algebra.)

More often than not, we will work with topological spaces that has infinitely many open sets. In this case it will be hard to explicitly list out all the open sets in the topological space. In this post, we will look at a topology call cocountable topology, and luckily, we can describe the open set of this topology, thus we do not need to describe a topological basis.

Given a ground set $X$, the cocountable topology $\tau$ on $X$ is the empty set together with the collection of subsets $U \subseteq X$ such that $X \setminus U$ is countable, namely the complement of $U$ is countable. For example, if $X = \{1,2,3\}$, then under the cocountable topology, none of the subsets of $X$ is open.

Now let’s assume that $X$ is an uncountable set.  By definition, $\emptyset, X \in \tau$. Then, let $\{U_{\lambda}\}_{\lambda \in \Lambda} \subseteq \tau$ be a collection of open sets in $X$ for some index set $\Lambda$. Thus $X \setminus U_{\lambda}$ is countable for all $\lambda \in \Lambda$. Now consider

$X \setminus \bigcup_{\lambda \in \Lambda} U_{\lambda} = \bigcap_{\lambda \in \Lambda} [X \setminus U_{\lambda}] \subseteq X \setminus U_{\lambda_0}$

by De Morgan’s law, where $\lambda_0 \in \Lambda$ is arbitrary. Since $X \setminus U_{\lambda}$ is countable for all $\lambda \in \Lambda$ and intersections will only result in smaller set, therefore $X \setminus \bigcup_{\lambda \in \Lambda} U_{\lambda}$ must be countable. Thus we may conclude that $\bigcup_{\lambda \in \Lambda} U_{\lambda} \in \tau$.

Finally, let $\{U_i\}_{i=1}^{n} \subseteq X$ be a finite collection. Then by De Morgan’s law, we have $X \setminus \bigcap_{i=1}^{n} U_i = \bigcup_{i=1}^{n} (X \setminus U_i)$. Notice that finite union of countable sets is still countable (in fact, countable union of countable sets is countable), thus $X \setminus \bigcap_{i=1}^{n} U_i$ is countable and $\bigcap_{i=1}^{n} U_i \in \tau$.

Therefore $\tau$ is indeed a topology.

# The Intersection of Topologies result in a Topology

Topology is the study of invariant properties of mathematical objects. Through the study of topology, mathematicians to translate properties between mathematical objects. There are many different subfields of topology, e.g. general topology (a.k.a. point-set topology), algebraic topology, etc. Unfortunately (or fortunately?), I have only encountered general topology in my undergraduate degree.

Perhaps one interesting (basic) result in topology is as follows

Given a space $X$ and a collection of topologies $\tau_i$, where $i \in I$ and $I$ is an index set, the intersection $\bigcap_{i \in I} \tau_i$ is also a topology on $X$.

Given a collection of objects that share property a property, let’s say property $a$, we do not expect the union and intersection of this collection to share the same properties. For instance, the set $\{n\}$, where $n$ is an integer, is finite. But when you take the union of all singleton set (namely $\{n\}$), we have an infinite set.

So why is the above statement true? To prove this, we need need at least the definition of a topology, so here it goes:

Let $X$ be a set, and $\mathcal{P}(X)$ be its power set. A subset $\tau \subseteq \mathcal{P}(X)$ is called a topology if the following conditions hold:

1. $\emptyset, X \in \tau$
2. Arbitrary union of elements of $\tau$ is also an element of $\tau$, i.e. $\bigcup_{\lambda \in \Lambda} U_{\lambda} \in \tau$ where $U_{\lambda} \in \tau$ is arbitrarily chosen.
3. Finite intersection of elements of $\tau$ is also an element of $\tau$, i.e. if $n \in \mathbb{Z}$, then $\bigcap_{i=1}^n U_{i} \in \tau$ where $U_{i} \in \tau$ is arbitrarily chosen

So far so good. Now that we know the definition of a topology, we can easily see that the union of topologies is NOT a topology; consider $X = \{a,b,c\}$, and we have two topologies $\tau_1 = \{\emptyset, \{a\}, X\}$ and $\tau_2 = \{\emptyset, \{b\}, X\}$, then the union is

$\tau_1 \cup \tau_2 = \{\emptyset, \{a\},\{b\},X\}$

This violates the arbitrary union property; $\{a\}$ and $\{b\}$ are elements of the union, but $\{a\} \cup \{b\} = \{a,b\}$ is not. Note that this is not to say that all unions of topologies does not result in a topology; it is only the counterexample to the claim that all unions of topologies gives a topology.

We have shown above that the union of topologies itself need not be a topology. So why is the intersection of of topologies is itself a topology? Clearly, by definition of topology, each topology includes the sets $\emptyset$ and $X$, thus the empty set $\emptyset$ and $X$ is in the intersection. Now let $I$ be an index set, and for $i \in I$, $U_{i} \in \bigcap_{\lambda \in \Lambda} \tau_{\lambda}$ be an arbitrary family of elements in the intersection of the topologies. Then by definition of topology, arbitrary unions of elements in $\{U_i\}_{i \in I}$ is also in $\tau_{\lambda}$ for all $\lambda \in \Lambda$. Thus the arbitrary union is also an element of $\bigcap_{\lambda \in \Lambda} \tau_{\lambda}$. By similar reasoning, we can also conclude that finite intersection of elements in $\bigcap_{\lambda \in \Lambda} \tau_{\lambda}$ is also an element of $\bigcap_{\lambda \in \Lambda} \tau_{\lambda}$ itself. Thus $\bigcap_{\lambda \in \Lambda} \tau_{\lambda}$ is a topology.

# Hypergraphs: Introduction

Graph theory is the study of graphs, which can be used to model relationships, routes, etc. People understands a lot about graphs. However, there is a generalisation of graph theory that we can talk about – the hypergraphs!
Like graph theory, hypergraphs contains vertices, denoted by $V$, and edges which we call hyperedges, denoting $E$. Instead of connecting two vertices $u,v \in V$, a hyperedge $e \in E$ connects any subset of $V$, i.e. $e$ is an element of the power set of the vertices.

Intuitively, hypergraphs are just a set of vertices, and hyperedges are lines that connects some vertices together. For instance, the $n \times n$ grid is a hyper graph, with $n^2$ vertices and $2n$ hyperedges. In the $n \times n$ grid, each pair of hyperedges, $e,e' \in E$ intersects at a vertex $v \in V$ uniquely. Another example of a hypergraph is the Fano plane, which is the hypergraph below

As we can see, there are seven vertices in total in the hypergraph above. There are also seven hyperedges in the hypergraph above, each hyperedge is represented by a unique colour itself.

We can also talk about the properties of a hypergraph just as we do in normal graphs. Recall that in a graph, we say that a graph is $k$-regular if each vertex has exactly degree $k$. Similarly, a hypergraph is $k$-regular if for each vertex in the hypergraph, the degree of the vertex (number of lines that pass through a vertex) is exactly degree $k$. So a $n \times n$-grid is 2-regular, and the Fano plane is 3-regular.

Another interesting property that we want to inherit from graph theory is the dual. Amazingly, basic properties of duality that we learn from normal graph theory is also true for hypergraphs. Perhaps the most important such property is that $G^{**} \cong G$ for any hypergraph $G$, where $G^*$ denotes the dual graph of the graph $G$. This is another post for another time.

There are many other types of hypergraphs, and they are used in many ways. Hopefully this post helps you with some intuition in hypergraphs.