Intuition behind definition of Connected Space in Topology

Question

There are 2 definitions of Connected Space in my lecture notes, I understand the first one but not the second. The first one is:

A topological space $(X,\mathcal{T})$ is connected if there does not exist $U,V\in\mathcal{T}$ such that $U\neq\emptyset$, $V\neq\emptyset$, $U\cap V=\emptyset$ and $X=U\cup V$

which makes sense. It is saying that connected spaces can't be cut up into parts that have nothing to do with eachother.

The second definition is:

A topological space $(X,\mathcal{T})$ is connected if $\emptyset$ and $X$ are the only subsets of $X$ which are closed and open

which makes no intuitive sense to me, especially as a definition of connectedness.

Any intuitive explanation behind this second definition?

Answer 1

First of all the definitions are equivalent, you already got a few answers about that. I'll try to add some intuition. If $X$ is a topological space and $A\subseteq X$ then you can split the space into three parts: the interior of $A$, the boundary of $A$ and the exterior of $A$. The set $A$ is open if its boundary is contained in $X\setminus A$, and it is closed if its boundary is contained in $A$. So to say that $A$ is both open and closed is the same thing as to say that the boundary of $A$ is empty. Now imagine that the interior and the exterior of such a set $A$ are both not empty. The set has no boundary which can be crossed so intuitively to get from a point in the interior to a point in the exterior we have to "jump". This exactly means that we split the space into two parts which have nothing to do with each other, so such a space $X$ is not connected.

On the other hand if every set $A\subseteq X$ such that both $A$ and $X\setminus A$ are not empty has a boundary then intuitively it means that we can enter any set and get out of any set in a straight way by crossing its boundary. So it makes sense to say such a space $X$ is connected.

Answer 2

The second definition is equivalent to the first: suppose there is a set $U$ which is neither $X$ nor $\emptyset$ and is both open and closed. Then $U^c$, the complement of $U$ in $X$ is both open and closed as well (since $U$ is open implies $U^c$ is closed and $U$ is closed implies $U^c$ is open). Thus $U$ and $U^c$ are two non-empty open sets, neither of which is the whole space, with $\emptyset$ intersection and whose union is the whole space.

I'm not sure there's a particular intuition here: rather this is a technical reformulation of the first definition that can be useful as a way of determining when a space in connected or not.

Answer 3

Hint: Consider the following definition, halfway between the first and second:

$$\mbox{A topological space } (X,\mathcal{T}) \mbox{ is connected if there does not exist } U\in\mathcal{T} \mbox{ such that } U\neq\emptyset, (X\setminus U)\neq\emptyset, (X\setminus U)\in\mathcal{T} $$

Can you see how this is equivalent to the first definition?

Can you see how this is equivalent to the second definition?

Answer 4

If $U$ is closed and open, then so is $V:=U^\complement$. So if such $U$ exists that is neither empty nor the whole space, we have $X=U\cup V$ with $U\cap V=\emptyset$, $U\ne\emptyset$, $V\ne\emptyset$.