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 topological space is sequential if (in a certain sense) you can do topology in it using only sequences instead of more general nets.
Sequential spaces are a kind of nice topological space.
A sequential topological space is a topological space $X$ such that a subset $A$ of $X$ is closed if (hence iff) it contains all the limit points of all sequences of points of $A$—or equivalently, such that $A$ is open if (hence iff) any sequence converging to a point of $A$ must eventually be in $A$.
Equivalently, a topological space is sequential iff it is a quotient space (in $Top$) of a metric space.
The category of sequential spaces is a coreflective subcategory of the category of all topological spaces.
The category of sequential spaces is a reflective subcategory of the category of subsequential spaces, much as $Top$ itself is a reflective subcategory of the category of all pseudotopological spaces.
The category of sequential spaces is cartesian closed. See also convenient category of topological spaces.
Everything above assumes excluded middle (and probably at least countable choice). Without that, it's hard to prove the existence of any nontrivial sequential spaces.
For example, to prove that the real line is sequential as a topological space, we must find, given a set $A$ and a point $x$ such that every sequence converging to $x$ is eventually in $A$, a positive real number $\delta$ such that ${]x - \delta, x + \delta[} \subseteq A$, and it's not clear how to construct that number from the data at hand. (One might consider various specific sequences that converge to $x$, such as $(x + 1/n)_n$ and $(x - 1/n)_n$, and use them to find upper bounds on $\delta$; but no finite set of sequences will give an entire interval around $x$, and proving that an infinite set of sequences that does cover an entire interval has a uniform positive upper bound on $\delta$ is very tricky.)
The usual proof that the real line (or any first-countable topological space) is sequential uses excluded middle and countable choice: Supposing that $A$ is not open, consider $x \in A$ such that $x \notin Int(A)$, pick for each $\delta = 1/n$ (or for each of the countably many basic neighbourhoods $U_n$ of $x$ in a general first-countable space) a point $y_n$ such that $d(x,y_n) \lt 1/n$ (or such that $y_n \in U_n$) but $y_n \notin A$ (which must exist since none of these balls/neighbourhoods are contained in $A$), note that $\lim_n y_n = x$, and get a contradiction.
For this reason, constructive analysis often requires the use of general nets (or filters) in situations where classical analysis can get by with sequences. (It is trivially true, in any topological space, that a set $A$ is open if every net that converges to an element $x$ of $A$ belongs eventually to $A$, or equivalently that $A$ belongs to any filter that converges to $x$; you just use the neighbourhood filter of $x$.)
properties | implications |
---|---|
metrisable: topology is induced by a metric | a metric space has a $\sigma$-locally discrete base |
second-countable: there is a countable base of the topology. | Nagata-Smirnov metrization theorem |
$\sigma$-locally discrete base, i.e. $X$ has a $\sigma$-locally discrete base. | a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$. |
$\sigma$-locally finite base, i.e. $X$ has a countably locally finite base. | second-countable spaces are separable: choose a point in each set of countable cover. |
separable: there is a countable dense subset. | second-countable spaces are Lindelöf |
Lindelöf: every open cover has a countable sub-cover. | weakly Lindelöf spaces with countably locally finite base are second countable |
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense. | separable metacompact spaces are Lindelöf |
countable choice: the natural numbers is a projective object in Set. | separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover. |
metacompact: every open cover has a point-finite open refinement. | Lindelöf spaces are trivially also weakly Lindelöf. |
first-countable: every point has a countable neighborhood base | a space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base. |
Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$ | a first-countable space is Fréchet-Urysohn: obvious |
sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$ | obviously, a Fréchet-Uryson space is sequential |
countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$. | obviously, a sequential space is countably tight. |
Last revised on April 5, 2019 at 19:42:32. See the history of this page for a list of all contributions to it.