What are some examples of open sets that are NOT neighborhoods?

Question

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:

1. The empty set and the carrier set belong to the topology. That is, $\varnothing \in \tau$ $\text{and } X \in \tau \text{.}$
2. Any arbitrary (finite or infinite) union of members of $\tau$ belongs $\text{to }\tau \text{.}$
3. 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.

Answer 1

An easy consequence of the definition is that an open set is a neighborhood of each of its members. Namely, if $U$ is open and $p \in U$, take $V = U$ in the definition of neighborhood.