Space Suggestion: Direct limit of finite-dimensional Euclidean spaces #1738

Space Suggestion

Let $X$ be the Direct limit of finite-dimensional Euclidean spaces.

$X$ is the subset of $\mathbb{R}^\omega$ consisting of all eventually $0$ sequences, and $X$ has the topology coinduced by the family of inclusions of Euclidean spaces $\mathbb{R}^n \hookrightarrow X$, $x \mapsto (x^1, \ldots, x^n, 0, \ldots)$. This means that $X$ has the finest topology such that each inclusion $\mathbb{R}^n \hookrightarrow X$ is continuous.

Equivalently, the set $U \subset X$ is open if and only if $U \cap \mathbb{R}^n$ is open in $\mathbb{R}^n$ for each $n$; here we identify each Euclidean space $\mathbb{R}^n$ with its image.

Equivalently, $X \subset \mathbb{R}^\omega$ has the subspace topology as a subspace of $\mathbb{R}^\omega$ with the box topology; see S107 and MSE. We mention that $X$ is a path component of $\mathbb{R}^\omega$, since it is a path-connected quasi-component; see MSE.

(Also sometimes defined as a topological vector space / abelian group with coproduct notation $\bigoplus_{n = 1}^\infty \mathbb{R}$.)

Notes on the above

We should discuss which terminology to use. Page 4 of Spanier's book, which might be dated now, uses the coinduced terminology. Final Topology on Wikipedia has some synonyms. Direct limits / colimits are a general concept. Lundell's The Topology of CW Complexes uses the weak topology terminology; see page 2 and Theorem 2.4 on page 47 . Paul Frost has two posts discussing the terminology of "weak topology"; post1 and the longer post2, which includes a more intricate looking description. On page 523, Hatcher uses the word "generated" for this concept (i.e., A topological space $X$ is said to be generated by a collection of subspaces ${X_\alpha}$ if $X = \bigcup_\alpha X_\alpha$ and a set $A \subset X$ is closed iff $A \cap X_\alpha$ is closed in $X_\alpha$ for each $\alpha$. Sakai introduces the concept of a direct limit of a tower of spaces for this. One distinction is that he uses an increasing sequence of subspaces ${X_n}$ and defines the space $\underrightarrow{\lim} X_n$ with underlying set $\bigcup X_n$.

The second definition more or less corresponds exactly to what Husemoller calls the $\Phi$-topology for a family $\Phi$ of spaces defined on subsets of a fixed set.

Milnor-Stasheff has the following blurb on page 63:

It is customary in algebraic topology to call this the "weak topology," a weak topology being one with many open sets. This usage is unfortunate since analysts use the term weak topology with precisely the opposite meaning. On the other hand the terms "fine topology" or "large topology" or "Whitehead topology" are certainly acceptable.

Rationale

This space came up while we were working on the box product $\mathbb{R}^\omega$. Including it simplifies at least one trait for that space (SLSC for $\mathbb{R}^\omega$ follows from contractibility of $X$, see #1735 (comment)). I hope it will simplify others and/or many of the traits of this one will be derivable from the fact that this is a subspace of $\mathbb{R}^\omega$. It also just seems like a natural space worth having on its own right.

The space $\mathbb{R}^\infty$ comes up when discussing universal bundles and classifying spaces. For example, a universal principal $G$-bundle for $G = O(n)$ is $V_n(\mathbb{R}^\infty) \to Gr_n(\mathbb{R}^\infty)$. Here $V_n(\mathbb{R}^\infty)$ is the infinite Stiefel manifold, which is the space of orthonormal $n$-frames in $\mathbb{R}^\infty$. And $Gr_n(\mathbb{R}^\infty)$ is the infinite Grassmannian, which is the space of all $n$-dimensional vector subspaces in $\mathbb{R}^\infty$.

References to this space

Defined in Milnor-Stasheff on pages 62-63, as "direct limit of the sequence $\mathbb{R}^1 \subset \mathbb{R}^2 \subset \cdots$. On page 63, Milnor-Stasheff defines the associated vector bundle to the principal $G$-bundle above, where the total space is defined as a subspace of $Gr_n(\mathbb{R}^\infty) \times \mathbb{R}^\infty$.

Husemoller clearly presents this space and related ones under Definition 1.1 on page 2.

Sakai briefly mentions $X$ in Remark 6 on page 56.

Has some mentions in Hatcher, but the definition is not very formal; I think Example 0.5 is the definition. It's tangentially involved in Example 1B.3 on page 88. Example 4.53 on page 381 discusses Stiefel and Grassmann manifolds. And there are some other appearances.

Listed in the Wikipedia article 'Fréchet–Urysohn space' under the subsection Direct limit of finite-dimensional Euclidean spaces.

Appears briefly in Munkres. It's in exercise 7 on page 118, exercise 5 on page 127, exercise 8(b) on page 128, and exercise 3(c) on page 275.

.. to be continued. I'm sure there are other good references.

Relationship to other spaces and properties

I came up with a list of some easy traits, without attempting to be non-redundant (pastable into Dan's tool, explained after):

P199
P87
P30
P67
P140
P56
~P23
P79
~P80
P17
~P204
P26
P223
P111

Embeds as a path component in Countable box product of reals S107.

$X$ admits a real topological vector space structure (not inherited from $\mathbb{R}^\omega$, since scalar multiplication on $\mathbb{R}^\omega$ is not continuous), hence is contractible P199 and has a group topology P87. It does inherit its topology and additive subgroup structure from $\mathbb{R}^\omega$. (Maybe it's worth mentioning that the quotient space $\mathbb{R}^\omega / \mathbb{R}^\infty$ is called the nabla product $\nabla^\omega \mathbb{R}$ in Williams' article Box products from Handbook for Set Theoretic Topology.)

$X$ admits a CW complex structure, hence is paracompact P30 (Lundell Theorem 4.2) , T6 P67 (Lundell Theorem 4.2 + Proposition 4.3), and compactly generated (k-space P140) (follows from the definition of CW complex). Also implies locally contractible P223 (Hatcher Proposition A.4).

• Some of these properties may be more direct without introducing CW complex language for now, since pi-base doesn't have this concept.

$X = \bigcup \mathbb{E}^n$ expresses $X$ as a countable union of nowhere dense closed sets, hence $X$ is Meager P56. It's also clear that $X$ is $\sigma$-compact P17 from this. Separable P26 is also clear.

Every compact subset of $\mathbb{R}^\infty$ is contained in some $\mathbb{R}^N$: Let $K \subset \mathbb{R}^\infty$ be a compact set. If $K \not\subset \mathbb{R}^n$ holds for all $n$, then one can construct a countably infinite subset $D \subset K$ such that $D \cap \mathbb{R}^n$ is finite for each $n$. This contradicts compactness of $K$ because it implies $D$ is a closed discrete subset of $K$. This fact is mentioned here #1735 (comment). One corollary is that $X$ is not weakly locally compact P23. Also, I think it's pretty easy to argue $X$ is hemicompact P111 using this.

The Wikipedia entry says $X$ is sequential P79 but not Frechet-Urysohn P80.

Constructing a piecewise defined path shows $X$ does not have a cut point P204.

The $\mathbb{R}^\infty$ defined above has a finer topology than $c_{00} \subset \ell^2(\mathbb{R})$ (with the norm topology), which is probably an interesting space pi-base might eventually include too; see S30.

Other useful stuff

Sakai's Proposition 2.8.3 has the useful fact that if $Y$ is locally compact then $(\underrightarrow{\lim} X_n) \times Y \cong \underrightarrow{\lim} (X_n \times Y)$; with this one easily shows scalar multiplication in $X$ is continuous.

Remarks

Some negative hereditary Countable box product of reals S107 traits might need to be moved over if they're also negative for $X$.