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 Jordan curve theorem is a basic fact in topology. It stands out as having a statement that intuitively seems completely obvious, but which turns out to require a non-trivial formal proof.
The theorem states that every continuous loop (where a loop is a closed curve) in the Euclidean plane which does not intersect itself (a Jordan curve) divides the plane into two disjoint subsets (the connected components of the curve's complement), a bounded region inside the curve, and an unbounded region outside of it, each of which has the original curve as its boundary.
Given a notion of real number $\mathbb{R}$ in constructive mathematics, a Jordan curve consists of a set $J$ with a specific injection $i:J \to \mathbb{R}^2$ into the Euclidean plane $\mathbb{R}^2$, and a homeomorphism $h:\mathbb{S}^1 \to \mathrm{im}(i)$ from the topological unit circle $\mathbb{S}^1$ to the image of the subset injection.
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section under construction
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In constructive mathematics, the Jordan curve theorem is stated as
Given a Jordan curve $J$, two points $a$ and $b$ can be constructed off of $J$, such that given any point $c$ off of $J$, we can construct a polygonal path that is bounded away from $J$, and that joins $c$ to one of $a$ or $b$, and any polygonal path joining $a$ and $b$ comes arbitrarily close to $J$.
In cohesive homotopy type theory, the Jordan curve theorem says that:
Let $J$ be a Jordan curve in the Euclidean plane $J \subseteq \mathbb{R}^2$ with injection $i:J \hookrightarrow \mathbb{R}^2$, and let $\mathbb{R}^2 \setminus J$ be the set of all points in $\mathbb{R}^2$ which is away from all points in the Jordan curve $J$.
The shape of $\mathbb{R}^2 \setminus J$ is equivalent to $\mathbb{1} + S^1$, where $\mathbb{1}$ is the unit type and $S^1$ is the circle type.
The Jordan–Brouwer separation theorem states that, in the Cartesian space $\mathbb{R}^n$, every simple closed continuous hypersurface (defined as the image of a continuous injection from the sphere $S^{n-1}$) has an exterior with two connected components, one bounded (the inside of the hypersurface) and one unbounded, each of which has the original hypersurface as its boundary. (Furthermore, the continuous map defining the hypersurface can be extended to an auto-homeomorphism of $\mathbb{R}^n$ under which the inside and outside appear as the images of the inside and outside of the sphere.)
G. Berg, W. Julian, R. Mines, F. Richman, The constructive Jordan curve theorem, Rocky Mountain J. Math. 5(2): 225-236 (Spring 1975). (DOI:10.1216/RMJ-1975-5-2-225)
Wikipedia, Jordan curve theorem
Last revised on December 14, 2022 at 02:57:51. See the history of this page for a list of all contributions to it.