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 K-topology is a topology on the set of real numbers $\mathbb{R}$ which is finer than the Euclidean metric topology, but such that the new open sets are ‘piled up’ on the positive side of 0 (and actually are all contained in the half-open interval $[0,1)$) and have lots of ‘holes’. It is useful for constructing counterexamples in topology.
For instance the K-topology is an example of a topological space that is Hausdorff $(T_2)$ but not normal ($T_4$).
Write
for the subset of natural fractions inside the real numbers.
Define a topological basis $\beta \subset P(\mathbb{R})$ on $\mathbb{R}$ consisting of all the open intervals as well as the complements of $K$ inside them:
The topology $\tau_{\beta} \subset P(\mathbb{R})$ which is generated from this topological basis is called the K-topology.
We may denote the resulting topological space by
The real numbers with their K-topology (def. 1) $\mathbb{R}_K$ are a Hausdorff topological space which is not a regular Hausdorff space (hence in particular not a normal Hausdorff space).
By construction the K-topology is finer than the usual euclidean metric topology. Since the latter is Hausdorff, so is $\mathbb{R}_K$. It remains to see that $\mathbb{R}_K$ contains a point and a disjoint closed subset such that do not have disjoint open neighbourhoods.
But this is the case essentially by construction: Observe that
is an open subset in $\mathbb{R}_K$, whence
is a closed subset of $\mathbb{R}_K$.
But every open neighbourhood of $\{0\}$ contains at least $(-\epsilon, \epsilon) \backslash K$ for some positive real number $\epsilon$. There exists then $n \in \mathbb{N}_{\geq 0}$ with $1/n \lt \epsilon$ and $1/n \in K$. An open neighbourhood of $K$ needs to contain an open interval around $1/n$, and hence will have non-trivial intersection with $(-\epsilon, \epsilon)$. Therefore $\{0\}$ and $K$ may not be separated by disjoint open neighbourhoods, and so $\mathbb{R}_K$ is not normal.
The set of rational numbers, with the subspace topology inherited from the inclusion $\mathbb{Q} \hookrightarrow \mathbb{R}_K$ into the real numbers with their K-topology, is an example of a non-regular, totally path-disconnected Hausdorff space. This space can be used to construct a quasitopological groupoid which isn’t a topological groupoid.