"Principal uniform space" vs "discrete uniform space"?

Question

Which terms are better for a uniform space such that the set of entourages is a principal filter?

"Principal uniform space" or "discrete uniform space"?

"Principal uniformity" or "discrete uniformity"?

Answer 1

For any 2 sets $A,B$, let's define $$[\![A,B]\!]=\{L\subseteq B\mid A\subseteq L \}$$

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Let $X$ be a set. For each nonempty $A\subseteq X$, $[\![A,X]\!]$ is filter on $X$. Any filter on $X$ in this form, is called a principal filter.

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Any uniformity $\mathcal D$ on a set $X$ of the form $[\![H,X^2]\!]$, where $H\subseteq X^2$, can be called a principal uniformity.

Assuming $\Delta_X\subseteq H$, where $\Delta_X$ is the diagonal, $[\![H,X^2]\!]$ is a uniformity on $X$ iff

1. $H=H^{-1}$.
2. $H=H\circ H$.

As examples:

• $[\![X^2,X^2]\!]=\{X^2\}$ is the trivial uniformity. (trivial as applied to the topology (not as a filter)).
• $[\![\Delta_X,X^2]\!]$ is the discrete uniformity (its topology is the discrete topology).

Note: Any filter of the form $[\![\{a\},X]\!]$ is called a trivial (ultra)filter.