11 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 7.2.2 | ||
| $\mathfrak{E}(\sigma, \mathfrak{U})$ | $\sigma$-uniformity, generated by $\{E(S,U) \mid S \in \sigma,\ U \in \mathfrak{U}\}$. | Definition 7.2.2 | ||
$\mathrm{supp}(f)$ | Support of $f$. | Definition 5.19.1 | ||
| $C_{c}(X; E)$ | Compactly supported continuous functions $X \to E$. | Definition 5.19.2 | ||
| $C_{c}^{+}(X)$ | Compactly supported continuous functions $X \to [0, \infty)$ | Definition 5.19.3 $f \prec U$ | $f \in C_{c}(X; [0,1])$ with $\mathrm{supp}(f) \subset U$. | Definition 5.19.4 |
| $C_{0}(X; E)$ | Continuous functions vanishing at infinity. | Definition 5.21.1 | ||
| $BC(X; E)$ | Bounded continuous functions $X \to E$. | Definition 12.12.3 | ||
| $B^{\infty}(X; E)$ | Bounded $E$-valued strongly measurable functions on $X$. | Definition 16.1.1 | ||
| $B^{\infty}(X)$ | Bounded $\complex$-valued Borel measurable functions on $X$. | Definition 16.1.1 | ||
| $\mathcal{L}^{p}(X; E)$, $\mathcal{L}^{p}(\mu; E)$, $\mathcal{L}^{p}(X, \cm, \mu; E)$ | $E$-valued $p$-integrable functions on $X$. | Definition 16.1.2 | ||
| $\norm{f}_{L^p(X; E)}$ | $L^{p}$ norm of $f$: $\braks{\int \norm{f}_E^p d\mu}^{1/p}$. | Definition 16.1.2 | ||
| $\mathcal{L}^{\infty}(X; E)$, $\mathcal{L}^{\infty}(\mu; E)$, $\mathcal{L}^{\infty}(X, \cm, \mu; E)$ | $E$-valued essentially bounded functions on $X$. | Definition 16.1.3 | ||
| $\norm{f}_{\mathcal{L}^\infty(X; E)}$ | Essential supremum of $f$. | Definition 16.1.3 | ||
| $L^{p}(X, \cm, \mu; E)$ | $E$-valued $L^{p}$ space on $(X,\cm,\mu)$; quotient of $\mathcal{L}^{p}$ by a.e.-equality. | Definition 16.1.8 | ||
$\mathscr{N}$ | The Baire space. | Definition 10.3.5 |
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