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
Let $X$ be a topological space and $\mathcal{U}$ an open cover thereof. A (continuous) path $\gamma \colon I \to X$ can pass through many of the elements of $\mathcal{U}$ as it winds its way around $X$. We can decompose that path into segments such that each segment lies wholly inside one of the open sets in $\mathcal{U}$. The Schedule Theorem says that this can be done continuously over all paths in $X$.
This was proved by Dyer and Eilenberg and applied to the question of fibrations over numerable spaces.
The idea of a schedule is that it is a way of decomposing a length into pieces and then assigning a label to each piece. This clearly fits with the stated purpose of these things since we wish to decompose a path into pieces and assign an open set to each piece.
To make this precise, we start with a set of labels. Following Dyer and Eilenberg, let us write this as $A$. Lengths are positive real numbers and so we also need the set of such, Dyer and Eilenberg denote this by $T$; thus $T \coloneqq \mathbb{R}_{\ge 0}$.
The schedule monoid of $A$ is the free monoid on the set $A \times T$. It is written $S A$. Its elements are schedules in $A$.
A schedule in $A$ is thus a finite ordered list of pairs $(a,t)$ where $a \in A$ and $t \in T$.
There are two notions of length for a schedule. There is the word length which simply counts the number of pairs. Then there is the function $l \colon S A \to T$ defined by $l((a_1,t_1) \cdots (a_k,t_k)) = t_1 + \cdots + t_k$. There is also a right action of $T$ on $S A$ which simply multiplies all of the lengths: $((a_1,t_1) \cdots (a_k,t_k)) \cdot t = ((a_1,t_1 t) \cdots (a_k,t_k t))$. Then $l(s t) = l(s) t$.
A schedule is said to be reduced if all of its terms, $(a,t)$, have non-zero length, i.e. $t \gt 0$. The set of reduced schedules forms a submonoid of $S A$ which is written $R S A$.
The empty schedule is reduced.
There is a retraction map $\rho \colon S A \to R S A$ defined by removing all terms with zero length part.
The schedule monoid is given a topology so that the labels are discrete and the lengths topologised as usual. More concretely, given a word $a_1 a_2 \dots a_k$ of elements in $A$, the set of schedules of the form $(a_1,t_1) (a_2,t_2) \cdots (a_k,t_k)$ is in bijection with $T^k$ and we make that bijection a homeomorphism. Then $S A$ is topologised by taking the coproduct over the set of words in $A$. The reduced schedule monoid is topologised as the quotient of this.
Let $X$ be a topological space. Let $P X$ denotes its Moore path space. Suppose that we have a family $\mathcal{U}$ of subsets of $X$ indexed by some set $A$. Then we consider a schedule in $A$ as giving an ordered list of these subsets together with the times to be spent in each. For a path in $X$, and a schedule of the appropriate length, then we can ask whether or not the path fits (or obeys) the schedule. We make that precise as follows.
Suppose that we have $\alpha \in P X$ and $s \in S A$, and suppose that $s = (a_1, t_1) \cdots (a_k,t_k)$. Then we say that $\alpha$ fits the schedule s, written $\alpha \Vert s$, if the following conditions hold:
Here, $l \colon P X \to T$ is the function that assigns to a Moore path its length. The schedule designates a decomposition of $[0,l]$ into subintervals with $t_i$ being the length of the $i$th subinterval. Then saying that $\alpha$ fits the schedule $s$ means that $\alpha$ spends the $i$th subinterval in the open set $U_{a_i}$.
We can now state the main theorem.
Let $X$ be a topological space. Let $\mathcal{U}$ be a locally finite open covering of $X$ by numerable open sets with indexing set $A$. Then there is a covering $\mathcal{F}$ of $P X$ by closed sets and a family of continuous functions $f \colon F \to S A$, indexed by $F \in \mathcal{F}$ such that:
The first condition is purely about the covering. Dyer and Eilenberg use the term local covering for a covering by closed sets with this property.
There exists a continuous function $h \colon P X \to R S A$ such that $\alpha \Vert h(\alpha)$ if $l(\alpha) \gt 0$ and $h(\alpha) = \Lambda$ if $l(\alpha) = 0$.
Here, $\Lambda \in R S A$ is the empty word.
The original motivation for the notion of schedules was to prove the globalisation theorem for (Hurewicz) fibrations.
Let $p \colon Y \to B$ be a continuous function. Suppose that $\mathcal{U}$ is a locally finite covering of $B$ by numerable open sets with the property that for each $U \in \mathcal{U}$ then the restriction $p_U \colon Y_U \to U$ is a fibration. Then $p$ is a fibration.
The link between the globalisation theorem and the schedule theorem is the characterisation of Hurewicz fibrations in terms of Hurewicz connections.
Let $X$ be a topological space. Let $\mathcal{U}$ be a locally finite open covering of $X$ by numerable open sets and indexing set $A$.
Let us write $A^*$ for the free monoid on $A$. Then there is a function $A^* \times T \to S A$ which takes $(a_1 a_2 \cdots a_k, t)$ to the schedule $(a_1,t/k)(a_2,t/k)\cdots (a_k,t/k)$. We say that a path $\alpha \in P X$ evenly fits $s \in A^*$, and write this as $\alpha \Vert_e s$, if it fits the schedule corresponding to $(s,l(\alpha))$.
We need an initial technical result.
There is a locally finite covering $\mathcal{W} = \{W_s \mid s \in A^*\}$ of $P X$ by numerable open sets such that for $\alpha \in W_s$ then $\alpha$ evenly fits the word $s$.
Let us explain why this is a reasonable result. Consider a path, $\alpha$, of length $l$. We pull back the cover $\mathcal{U}$ to a cover of $[0,l]$. Using compactness of $[0,l]$ we can replace the pull-back cover by a finite family of open subintervals of $[0,l]$ which cover $[0,l]$. Each subinterval is labelled by an element of $\mathcal{U}$ (though a label might be reused). As the family is finite, the intersections are finite and therefore have a minimum length. Choose $n$ big enough so that $l/n$ is less than this minimum length. Then consider the subdivision of $[0,l]$ given by $\{0,1/n,\dots,l/n\}$. Our conditions on $n$ guarantee that every intersection of subintervals contains at least one of these division points. We can therefore assign to each subinterval of the form $[k/n, (k+1)/n]$ one of the original family of subintervals that contains it. Then we can assign the corresponding element of $\mathcal{U}$. Thus $\alpha$ fits evenly the corresponding word.
Thus the sets $Y_s \coloneqq \{\alpha : \alpha \|_e s\}$ cover $P X$. That they are open follows from the fact that the condition for membership depends on certain compact sets lying in certain open sets and we use the compact-open topology on $P X$.
What is more complicated is reducing the family to a locally finite one.
As $\mathcal{W}$ is locally finite and its elements are numerable, we can choose a numeration that is also a partition of unity. That is, we can choose continuous functions $q_s \colon X \to [0,1]$ with the property that $q_s^{-1}((0,1]) = W_s$ and $\sum_s q_s = 1$.
Let $\mathcal{B}$ be the set of finite subsets of $A^* \setminus \Lambda$ (where $\Lambda$ is the empty word). For $b \in \mathcal{B}$ we define
This is a covering of $P X$ by closed sets. As $\mathcal{W}$ is locally finite, for $\alpha \in P X$ there is some neighbourhood $V$ which meets only a finite number of the $\mathcal{W}$. These are indexed by elements of $A^*$, indeed of $A^* \setminus \Lambda$, and so the set of indices is an element, say $b$, of$\mathcal{B}$. Then for $s \notin b$, $q_s \mid V = 0$ and so for $\beta \in V$, $\sum_{s \in b} q_s(\beta) = 1$, whence $V \subseteq D_b$. Thus each $\alpha$ is contained in the interior of some $D_b$.
Now let us put a total ordering on $A^*$. This induces a total ordering on each $b \in \mathcal{B}$ and thus allows us to define the partial sums of the summation $\sum_{s \in b} q_s$. Write these as $Q_i$, with $Q_0$ as the zero function.
Fix $b \in \mathcal{B}$ and write it as $b = \{s_1,s_2,\dots,s_k\}$ in the inherited ordering. Let $e = (l_1,r_1,\dots,l_k,r_k)$ be a list of integers with the property that $1 \le l_i \le r_i \le \#s_i$ where $\#s_i$ is the word length of $s_i$. Define:
This is closed in $D_b$ and the collection $\{D_{(b,e)}\}$ is a finite cover of $D_b$. The family $\{D_{(b,e)}\}$ ranging over all $b \in \mathcal{B}$ and suitable $e$ is the family $\mathcal{F}$ that we are looking for. It has the required covering property since the interiors of the $D_b$ cover $P X$.
Define $f_{(b,e)} \colon D_{(b,e)} \to S A$ as follows:
where $\sigma_i$ is the schedule with $\# \sigma_i = r_i - l_i + 1$ and $l(\sigma_i) = q_{s_i}(\alpha)$, and if $s_i = a_1 \cdots a_n$ then if $l_i \lt r_i$ we have
otherwise, $\sigma_i = (a_{l_i}, Q_i(\alpha) - Q_{i-1}(\alpha))$.
This is continuous and for $\alpha \in D_{(b,e)}$ then $\alpha$ fits $f_{(b,e)}(\alpha)$. Moreover, for $\alpha \in D_{(b,e)} \cap D_{(b',e')}$ then $\rho f_{(b,e)}(\alpha) = \rho f_{(b',e')}(\alpha)$.
Dyer, E. and E., Samuel. (1988). Globalizing fibrations by schedules. Fund. Math., 130, 125β136. MR0963792
Dyer, Eilenberg, MR0963792