topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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
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constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
basic constructions:
strong axioms
further
The principle of enough functions states that the formal locale of the function space of continuous real endofunctions is a spatial locale.
The principle of enough functions is true in classical mathematics, but cannot be proven in neutral constructive mathematics and so has to be added as an axiom.
Heine-Borel theorem (The locale of the unit interval is a spatial locale.)
fan theorem (The locale of Cantor space is a spatial locale.)
monotone bar induction (The locale of the Baire space of sequences is a spatial locale.)
Created on April 14, 2025 at 21:27:45. See the history of this page for a list of all contributions to it.