Definitions
Since "neighborhoods" can be defined differently (as noted in the comments to this question), here are the relevant definitions I'm working with:
Topology (defined via open sets)
(from wiki; slightly modified for clarity)
A topology on a set $X$ may be defined as a collection $\tau$ of subsets of $X\text{,}$ satisfying the following axioms:
- The empty set and the carrier set belong to the topology. That is, $\varnothing \in \tau$ $\text{and } X \in \tau \text{.}$
- Any arbitrary (finite or infinite) union of members of $\tau$ belongs $\text{to }\tau \text{.}$
- The intersection of any finite number of members of $\tau$ belongs $\text{to }\tau \text{.}$
The carrier set, along with its topology, is called a topological space and is denoted $\text{by }(X, \tau)\text{.}$
Open set
Any element in the topology is called an open set. That is, any set $U$ $\text{where }U \in \tau \text{.}$
Neighbourhood (of a point)
(from wiki; slightly modified for clarity)
If $(X, \tau)$ is a topological space and $p$ is a point in $X$, then a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ $\text{containing }p$,
$$p \in U \subseteq V \subseteq X\text{.}$$
The Question
What are some examples of open sets that are not neighborhoods? The only one that comes to my mind is this one:
- The empty set $\varnothing$ since it's open and contains no points of $X$, regardless of the topology.
I tried thinking about the discrete topology on $X\text{,}$ and briefly believed that singleton sets were an example (until I started writing out my train of logic on this post). My first line of thinking was something like this:
Every point $p \in X$ has a corresponding singleton set $\{p\} \in \tau$ (so it's open). However, $\{p\}$ is the smallest open set containing the point $p\text{,}$ so there doesn't exist any $U$ so that $$p \in U \subseteq \{p\} \subseteq X\text{.}$$
But then I remembered that any set is a subset of itself, so $\{p\} \subseteq \{p\}$ made me realize that the sets $U$ and $V$ in the definition of neighborhood could be the same set.
I guess what I'm really wondering is whether there are non-empty open sets (in some topology) that are not neighborhoods.