9 Notations

Notation Description Source
$\mathcal{N}_{X}(A)$, $\mathcal{N}(A)$, $\mathcal{N}^{o}(A)$ Neighbourhood filter at $A$; open neighbourhoods of $A$. Definition 5.4.1
$C(X; Y)$ Continuous functions $X \to Y$. Definition 5.6.1
$E(d, r)$ $\{(x,y) \in X \times X \mid d(x,y) < r\}$ for pseudometric $d$. Definition 6.3.3
$B(x, r)$ Open ball $\{y \in X \mid d(x,y) < r\}$ for pseudometric $d$. Definition 6.3.3
$B(A, \varepsilon)$ $\varepsilon$-fattening $\{x \in X \mid d(x, A) < \varepsilon\}$ of $A$. Definition 8.2.3

$UC(X; Y)$
Uniformly continuous functions $X \to Y$. Definition 6.2.1
$U^{-1}$ Inversion of $U \subset X \times X$. Definition 6.1.1
$U \circ V$ Composition of $U, V \subset X \times X$. Definition 6.1.2
$U(A)$ Slice of $U \subset X \times Y$ at $A \subset X$: $\{y \mid \exists x \in A,\, (x,y) \in U\}$. Definition 6.1.3
$E(S, U)$ Entourage of the form $\{(f,g) \in X^{T} \mid (f(x),g(x)) \in U\ \forall x \in S\}$. Definition 7.1.2
$\mathfrak{E}(\mathfrak{S}, \mathfrak{U})$ $\mathfrak{S}$-uniformity, generated by $\{E(S,U) \mid S \in \mathfrak{S},\ U \in \mathfrak{U}\}$. Definition 7.1.2

$\mathrm{supp}(f)$
Support of $f$. Definition 5.19.1
$C_{c}(X; E)$ Compactly supported continuous functions $X \to E$. Definition 5.19.2
$f \prec U$ $f \in C_{c}(X; [0,1])$ with $\mathrm{supp}(f) \subset U$. Definition 5.19.3
$C_{0}(X; E)$ Continuous functions vanishing at infinity. Definition 5.21.1
$BC(X; E)$ Bounded continuous functions $X \to E$. Definition 10.11.3