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
The Sierpiński space $\Sigma$ is the topological space
whose underlying set has two elements, say $\{0,1\}$,
whose set of open subsets is $\left\{ \emptyset, \{1\}, \{0,1\} \right\}$.
(We could exchange “0” and “1” here, the result would of course be homeomorphic).
Equivalently we may think of the underlying set as the set of of classical truth values $\{\bot, \top\}$, equipped with the specialization topology, in which $\{\bot\}$ is closed and $\{\top\}$ is an open but not conversely.
In constructive mathematics, it is important that $\{\top\}$ be open (and $\{\bot\}$ closed), rather than the other way around. Indeed, the general definition (since we can't assume that every element is either $\top$ or $\bot$) is that a subset $P$ of $\Sigma$ is open as long as it is upward closed: $p \Rightarrow q$ and $p \in P$ imply that $q \in P$. The ability to place a topology on $\Top(X,\Sigma)$ is fundamental to abstract Stone duality, a constructive approach to general topology.
The Sierpiński space $\Sigma$ is the initial $\sigma$-frame.
In type theory, as a higher inductive type Sierpinski space $\Sigma$ is generated by the following constructors:
The Sierpinski space $\Sigma$ is inductively generated by
and the partial order type family $\leq$ is simultaneously inductively generated by
a family of dependent terms
representing that each type $a \leq b$ is a proposition.
a family of dependent terms
representing the reflexive property of $\leq$.
a family of dependent terms
representing the transitive property of $\leq$.
a family of dependent terms
representing the anti-symmetric property of $\leq$.
a family of dependent terms
representing that $\bot$ is initial in the poset.
three families of dependent terms
representing that $\vee$ is a coproduct in the poset.
two families of dependent terms
representing that $\Vee$ is a denumerable/countable coproduct in the poset.
a family of dependent terms
representing that $\top$ is terminal in the poset.
three families of dependent terms
representing that $\wedge$ is a product in the poset.
a family of dependent terms
representing the countably infinitary distributive property.
This Sierpinski space
is a contractible space;
has a focal point;
is a sober topological space (but not T1).
According properties are inherited by the Sierpinski topos and the Sierpinski (∞,1)-topos over $Sierp$.
The Sierpinski space $S$ is a classifier for open subspaces of a topological space $X$ in that for any open subspace $A$ of $X$ there is a unique continuous function $\chi_A: X \to S$ such that $A = \chi_A^{-1}(\top)$.
Dually, it classifies closed subsets in that any closed subspace $A$ is $\chi_A^{-1}(\bot)$. Note that the closed subsets and open subsets of $X$ are related by a bijection through complementation; one gets a topology on the set of either by identifying the set with $\Top(X,\Sigma)$ for a suitable function space topology. (This part does not work as well in constructive mathematics.)
In synthetic topology, an analogue of the Sierpinski space is called a dominance.
Wikipedia, Sierpinski space
Paul Taylor, Foundations for computable topology – 7 The Sierpinski space (html)
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type (abs:1610.09254)
Martin Escardó, Cory Knapp, Partial Elements and Recursion via Dominances in Univalent Type Theory (pdf)
Andrej Bauer, Davorin Lešnik, Metric spaces in synthetic topology (pdf)
Last revised on June 9, 2022 at 23:02:22. See the history of this page for a list of all contributions to it.