a counterexample for Uniform Spaces

Question

Uniform Space is a generalization of metric spaces .

In a uniform space the closure of a singleton $\{x\}$ is the intersection of all neighborhoods of $x$.

Find an infinite topological space such that

• it is not a uniform space.

• the closure of any singleton $\{x\}$ is the intersection of all neighborhoods of $x$.

Answer 1

HINT: A topological space is uniformizable if and only if it is completely regular, so you need to find a space that is not completely regular. On the other hand, if $X$ is a Hausdorff space and $x\in X$, then $\{x\}$ is the intersection of the neighborhoods of $x$. (Why?) Thus, it suffices to find a Hausdorff space that is not completely regular, and there are many examples of those; references to two are given in this answer. This answer has an example of a Hausdorff space that is not even regular.