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
Georg Cantor used to define the continuum as a perfect space that is connected as well. Hence, the property of ‘being perfect’, as the name indicates as well, can be viewed as forming part of the concept of a ‘prototypical’ topological space.
A topological space $X$ is perfect if it has no isolated points, i.e., if every point $x$ belongs to the topological closure of its complement $X \setminus \{x\}$. Sometimes one requires a perfect space to be inhabited (nonempty), although it is better to allow the empty space.
In a topological space $X$, a subset is said to be perfect if it is closed in $X$ and perfect in its subspace topology. In other words, a set $A$ is perfect if and only if it equals its set $A'$ of accumulation points. Because of the closure requirement, being a perfect set/subset/subspace depends on the ambient space and is stronger than being a perfect space. (But $X$ is a perfect subset of itself iff $X$ is a perfect space.)
In a topological space $X$, a subset $A$ has the perfect-set property if it is either countable (possibly finite or even empty) or has an inhabited perfect subset. Of course, any perfect set has the perfect-set property.
If for each set $B$, ($B$ has a point apart from each point in $A$ if $B$ is inhabited and $B$ is perfect) and ($B$ is empty if $B$ is contained in $A$ and $B$ is perfect) and ($B$ has an isolated point if $B$ is contained in $A$ and inhabited), then $A$ is countable.
The empty space is perfect (unless it is excluded by fiat).
The Cantor space $2^\mathbb{N}$ is perfect, in fact, it is the only totally disconnected, compact, metrizable space that is perfect.
The real line is perfect.
Every Polish space (including the previous two examples) is perfect. Furthermore, any closed subset of a Polish space has the perfect-set property.
Using the axiom of choice, one may prove the existence of subsets of the real line that do not have the perfect-set property. In contrast, one of the axioms of dream mathematics is that every subset of the real line does have this property. This makes the continuum hypothesis a theorem of dream mathematics.
Every topological space $X$ is the disjoint union of a scattered subset and a perfect subset.
Every perfect subset of a Polish space $X$ (including $X$ itself) has the cardinality of the continuum? ($\beth_1$). (This is why the continuum hypothesis follows from the non-classical axiom that every subset of the real line has the perfect-set property.)
Every closed subset of a Polish space is a unique disjoint union of a countable set and a (possibly empty) perfect set.
Last revised on June 28, 2020 at 09:16:11. See the history of this page for a list of all contributions to it.