**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

Given a set $T$ with a dense linear order $\lt$, a pair of subsets $(L, R)$ of $T$ with injections $i_L:L \to T$ and $i_R:R \to T$ is a **Dedekind cut structure** if it comes equipped with the following structure

- an element $l \in L$
- an element $r \in R$
- for every element $a \in L$ and $b \in T$, a function$c_d(a, b):]b, a[ \to \{c \in L \vert i_L(c) = b\}$
- for every element $a \in R$ and $b \in T$, a function$c_u(a, b):]a, b[ \to \{c \in R \vert i_R(c) = b\}$
- for every element $a \in L$, an element$o_d(a) \in \{b \in L \vert i_L(a) \lt i_L(b)\}$
- for every element $a \in R$, an element$o_u(a) \in \{b \in R \vert i_R(b) \lt i_R(a)\}$
- for every element $a \in L$ and $b \in R$, an element$t(a, b) \in ]i_L(a), i_R(b)[$
- for every element $a \in T$ and $b \in T$, a function$L(a, b):]a,b[ \to (\{c \in L \vert i_L(c) = a\} \uplus \{c \in R \vert i_R(c) = b\})$

where $]a, b[$ is the open interval bounded by $a$ from below and by $b$ from above.

- Auke Booij,
*Extensional constructive real analysis via locators*, (abs:1805.06781)

Last revised on June 10, 2022 at 10:58:11. See the history of this page for a list of all contributions to it.