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
A fundamental theorem of topology: for $n_1, n_2 \in \mathbb{N}$ two different natural numbers, $n_1 \neq n_2$, then the Cartesian spaces $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ of dimension $n_1$ and $n_2$, respectively, are not homeomorphic (while for $n_1 = n_2$ then of course they are). Hence dimension of topological manifolds is a topological invariant.
This seems intuitively obvious, but the formal proof (due to Brouwer around 1910) is comparatively hard, and the problem was an important open problem in the 19th century. To appreciate that the problem is harder than it may superficially seem it serves to notice that for every $n \in \mathbb{N}$ there are surjective continuous maps $\mathbb{R}^1 \to \mathbb{R}^n$ (the Peano curves), in addition to the obvious injective continuous maps $\mathbb{R}^1 \to \mathbb{R}^n$ for $n \geq 1$ (or indeed $\mathbb{R}^m \to \mathbb{R}^n$ for $m \leq n$), because of which there are also bijective (but discontinuous) maps $\mathbb{R}^m \to \mathbb{R}^n$ for $m, n \geq 1$ (by the Schroeder–Bernstein theorem).
(topological invariance of dimension)
For $n_1, n_2 \in \mathbb{N}$ then the Euclidean space $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ (regarded as topological spaces with their metric topology) are homeomorphic if and only if $n_1 = n_2$.
More generally:
(Brouwer invariance of domain theorem)
For $U \subset \mathbb{R}^n$ an open subset of a Cartesian space, for any $n \in \mathbb{N}$, and for $f \colon U \to \mathbb{R}^n$ a continuous injective map, then its image $f(U)$ is also an open subset.
In other words, continuous injections between Cartesian spaces of the same dimension are open maps.
This implies theorem 1:
For $n_1, n_2 \in \mathbb{N}$, there is a continuous injection from $\mathbb{R}^{n_1}$ to $\mathbb{R}^{n_2}$ if and only if $n_1 \leq n_2$.
If $n_1 \leq n_2$, use $\iota_{n_1,n_2}\colon (x_1, \ldots, x_{n_1}) \mapsto (x_1, \ldots, x_{n_1}, \vec{0})$, where $\vec{0}$ consists of $n_2 - n_1$ copies of $0$. If $n_1 \gt n_2$, then supposing a continuous injection $f\colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$, compose $f$ with $\iota_{n_2,n_1}$ to get a map from $\mathbb{R}^{n_1}$ to itself, also a continuous injection. By invariance of domain (theorem 2), the image of this map is open in $\mathbb{R}^{n_1}$, yet contained in the range of $\iota_{n_2,n_1}$, and the only open subset of that range is empty, a contradiction since $\mathbb{R}^{n_1}$ is not empty.
A homeomorphism is a continuous injection both ways, so $n_1 \leq n_2$ and $n_2 \leq n_1$ by lemma 1.
This only-if half of this argument immediately generalizes to any inhabited open subsets of $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$.
We discuss various proofs of the topological invariance of dimension (theorem 1).
of theorem 1
Consider relative ordinary cohomology $H_\bullet$ with coefficients in, says, the integers $\mathbb{Z}$. Recall that for $A \hookrightarrow X$ an inclusion of CW-complexes, then there is a long exact sequence of the form
Applied to the point inclusion $\{0\} \hookrightarrow \mathbb{R}^n$ and using that
$H^k(\mathbb{R}^n) = \left\{ \array{ \mathbb{Z} & \vert\, \text{if}\, k = 0 \\ 0 & \vert\, \text{otherwise}} \right.$
because $\mathbb{R}^n$ is homotopy equivalent to the point;
$H^k(\mathbb{R}^n \backslash \{0\}) = \left\{ \array{ \mathbb{Z} &\vert\, \text{if} \, k \in \{0,n-1\} \\ 0 & |\, \text{otherwise} } \right.$
because $\mathbb{R}^n \backslash \{0\}$ is homotopy equvaent to th (n-1)-sphere $S^{n-1}$
this gives for $n \geq 2$ that $H^k(\mathbb{R}^n , \mathbb{R}^n - \{0\}) = \left\{ \array{ \mathbb{Z} &\vert\, \text{if} \, k \in \{0,n\} \\ 0 & \vert\, \text{otherwise} }\right.$
Hence $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ do not have the same relative cohomology unless $n_1 = n_2$. Since every homeomorphism induces an isomorphism on (relative) cohomology, they cannot be homeomorphic (using excluded middle here).
A proof also follows from the properties of the Adams operations on topological K-theory, see for instance (Wirthmuller 12, p. 46).
The first proof is due to Brouwer around 1910.
Terry Tao, Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem
Wikipedia, Invariance of domain
A proof also follows from the properties of the Adams operations on topological K-theory, see for instance
Last revised on August 7, 2017 at 06:26:49. See the history of this page for a list of all contributions to it.