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

## 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.

• I think the OP's confusion stems from trying to make sense of a concept of "neighbourhood" as something distinct from "open set", when in reality the only meaningful use of the word neighbourhood is when specifying "neighbourhood of a particular point p." Jul 5 at 18:16
• Slight nitpick: one can also have a "neighbourhood of a set" en.wikipedia.org/wiki/… But you made me realize that I shouldn't have left it implicit for most of the post (figured the section title "Neighbourhood (of a point)" was enough 🤷). A part of my confusion likely stems from some mathematicians using "U is a neighborhood of a point p" to mean "U is an open set containing p." (p. 96 of Munkres Topology 2nd edition) But I think you're also correct that another part of my confusion was from not focusing on a point p. Jul 6 at 5:00

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.