Confusion about topology on CW complex: weak or final?

The topology of the CW complex is defined to be the weak topology: given the sequence of inclusions of the skeleta $X_0 \subseteq X_1 \subseteq_ \cdots$ a subset $A \subseteq X = \cup X_i$ is open iff $A \cap X_i$ is open for all $i$.

Nlab says this is equivalent to the colimit of the inclusions, and so the weak topology is the same as the final topology:

"W = “weak topology”: Since a CW-complex is a colimit in Top over its cells, and as such equipped with the final topology of the cell inclusion maps..."

(Remark 1.1, https://ncatlab.org/nlab/show/CW+complex)

Following the link there to "final topology" it then says the weak topology means the same as initial topology and the final topology is synonymous with the strong topology, i.e., the opposite of the weak topology... So which is correct, the weak or the strong? It seems that it cannot be both.

Solution 1:

We have inclusions $i_k : X_k \to X$. The skeleta $X_i$ have topologies, and with respect to these topologies each $X_i$ is a subspace of $X_{i+1}$.

The CW-complex $X$ is then endowed with the final topology with respect to the family $(i_k)$, i.e. with the finest topology such that all $i_k$ become continuous.

So why is it called the weak topology which in modern terms would in fact correspond to the concept of an initial topology?

It seems to me that in older literature the word "weak topology" was used as follows:

Given a set $X$ and a family $\{ X_\iota \}_{\iota \in I}$ of subsets of $X$ with each $X_\iota$ having a topology and assume that

a) $X_\iota \cap X_{\iota'}$ inherits the same subspace topology from $X_\iota$ and from $X_{\iota'}$

b) Either each $X_\iota \cap X_{\iota'}$ is open in both $X_\iota$ and $X_{\iota'}$ or each $X_\iota \cap X_{\iota'}$ is closed in both $X_\iota$ and $X_{\iota'}$

Then the weak topology on $X$ induced by $\{ X_\iota \}_{\iota \in I}$ consists of all $U \subset X$ such that each $U \cap X_\iota$ is open in $X_\iota$ (equivalently one can define closed sets $A$ by requiring $A \cap X_\iota$ closed in $X_\iota$).

See for example

Dugundji, James. "Topology Allyn and Bacon." Inc., Boston 5 (1966).

In modern language this topology would be called the final topology induced by the inclusions $X_\iota \to X$.

Added:

The first occurrence of the phrase "weak topology" I could find was in

Whitehead, John HC. "Combinatorial homotopy. I." Bulletin of the American Mathematical Society 55.3 (1949): 213-245.

In this paper the concept of "CW-complex" was introduced. However, it goes back to

Whitehead, John Henry Constantine. "Simplicial Spaces, Nuclei and m‐Groups." Proceedings of the London mathematical society 2.1 (1939): 243-327.

On p.316 the concept of a "topological polyhedron" was introduced. Such a space is given the weak topology with respect to its closed cells (although the word "weak topology" was not used).

In a more general context weak topologies have been considered for example in

Morita, Kiiti. "On spaces having the weak topology with respect to closed coverings." Proceedings of the Japan Academy 29.10 (1953): 537-543.

Cohen, D. E. "Spaces with weak topology." The Quarterly Journal of Mathematics 5.1 (1954): 77-80.

See also p.44 of

Hart, Klaas Pieter, Jun-iti Nagata, and Jerry E. Vaughan. Encyclopedia of general topology. Elsevier, 2003.

All these sources do not really clarify why the word "weak" was used. Intuitively one would expect that a weak topology on a set has fewer open sets (i.e. is coarser) than other possible topologies. For example, on polyhedra the weak topology (= CW topology) is in general finer then the metric topology and thus one could regard it as stronger than the metric topology.

The usage seems to have again historical reasons. In

Arens, Richard F. "A topology for spaces of transformations." Annals of Mathematics (1946): 480-495.

I found the following in section 3 "Comparison of topologies":

Suppose that $t$ and $t^*$ are two topologies for the same class of elements. If the open sets of $t$ are also open in $t^*$, we shall write $t \subset t^*$, and say that $t$ is stronger than $t^*$, and $t^*$ is weaker than $t$.

We see that weaker is the same as finer is modern language. However, there is a footnote saying

In our use of "strong" and "weak", we concur with Alexandroff and Hopf (1, p. 62), but not all writers. "Stronger", above, means roughly "more limit points".

The quoted book by Alexandroff and Hopf (in German) is from 1935. It is still available:

Alexandroff, Paul, and Heinz Hopf. Topologie I: Erster Band. Grundbegriffe der Mengentheoretischen Topologie Topologie der Komplexe· Topologische Invarianzsätze und Anschliessende Begriffsbildungen· Verschlingungen im n-Dimensionalen Euklidischen Raum Stetige Abbildungen von Polyedern. Springer-Verlag, 2013.

In this book the concept of a topological space is introduced as a set $X$ with (in modern language) a closure operator for subsets $M \subset X$ ($M \mapsto \overline{M}$). The authors use the word topological assignment to denote a closure operator.

This is no longer the standard approach, but it is equivalent to defining a topological space to be a set with a topology (system of open subsets).

For those who are historically interested let me give a translation of the relevant definition on p.62:

If two topological spaces $X_1, X_2$ have the same underlying set, and if the identity map $X_1 \to X_2$ is continuous, we say that the topological assignment on $X_1$ is at most as strong as that on $X_2$.

A footnote says

This denotation is justified: If the topological assignment on $X_1$ is at most as strong as that on $X_2$, then this means: If $p$ is a limit point of $M$ in $X_1$, then it is certainly a limit point of $M$ in $X_2$; but $p$ may be a limit point of $M$ in $X_2$ without being a limit point of $M$ in $X_1$. Thus the topological assignment on $X_2$ is a strengthening of the topological assignment on $X_1$.

Although the words "stronger" and "weaker" are not explicitiy used here, it is obvious that if the topology (= topological assignment) on $X_1$ is at most as strong as that on $X_2$, then the topology on $X_2$ is regarded as at least as strong as that of $X_1$, in other words as stronger. Thus the topology on $X_1$ is weaker than that on $X_2$. In terms of closure operators it means that $\overline{M}^{1} \subset \overline{M}^{2}$ for all subsets $M$. Whether the second closure operator should then be regarded as a strenghtening of the first is a philosophical question. But certainly one can argue that it "stronger absorbs points" than the first.