pi-base
Space Suggestion: sigma-product of countably many copies of R
Space Suggestion
$X$ is the Countable $\sigma$-product $\sigma(\mathbb{R}^\omega)$. I.e., $X$ is the subspace of finitely supported real-valued functions on $\omega$. I.e., $X$ is the subspace of $\mathbb{R}^\omega$ consisting of eventually zero sequences.
I believe $X$ is homeomorphic to the following spaces.
- The topological union $\bigcup_{n = 0}^\infty \mathbb{E}^n$, where $\mathbb{E}^n$ denotes $n$-dimensional Euclidean space. (I've seen topology on a direct limit coinduced by the functions $\mathbb{E}^n \to \lim_{n \to \infty} \{\mathbb{E}^n\}$, e.g. on page 5 of Spanier, but not topological union. It seems like a nice name for this, but I haven't found a proper reference yet.)
- The subspace of $\ell^2$ consisting of eventually zero sequences. Note that this space is implicit on line 2 of the second list of spaces in S30, and appears in Bing's paper referenced there. It could be nice to link S30 and this space. I kind of want to ask on MSE if $\ell^2 \backslash X'$ is homeomorphic to $\ell^2$ for any embedding $X'$ of $X$, but I'm not sure when I will since it might be a silly question.
For a homeomorphism of $X$ with the first space in this list, or of the first and second spaces, I believe it's simply a matter of applying the universal property of a direct limit, but this is unfamiliar territory for me so we should be careful.
Compare with S000181
Rationale
The $\sigma$-product in general is defined on page 40 of Encyclopedia of General Topology.
I first learned of this space from Exercise 7 on page 118 of Munkres, where it appears as $\mathbb{R}^\infty$, the subspace of $\mathbb{R}^\omega$ consisting of eventually zero sequences (appears with different topologies elsewhere in the book).
Relationship to other spaces and properties
Each $\mathbb{E}^n := \{x_i = 0, i > n\} \subset X$ is a closed subset of $X$ which has empty interior. Since $X = \bigcup_{n = 0}^\infty \mathbb{E}^n$, it follows that $X$ is not Baire. This distinguishes $X$ from S30, and for me is the justification to include it. The next challenge is to distinguish $X$ from the weak topology on $\ell^2$ S21, which is because S21 is not Metrizable but $X$ is metrizable as it is a subspace of a metrizable space, since S30 is metrizable. An explicit inherited metric is also clear in the $\ell^2$ model. Then $X$ will fill in the empty search:
π-Base, Search for ~baire + metrizable + connected
- It's contractible, by considering the straight line homotopy $x \mapsto (1-t) x$.
- It's ~Embeddable into Euclidean space P184, by pjzp's argument https://github.com/pi-base/data/issues/1029#issue-2724836928
Isn't this space what is called the $\sigma$-product of countably many copies of $\mathbb R$? (That's "sigma-product" with a lowecase sigma, as opposed to $\Sigma$-product with an uppercase sigma, which is a different thing, which would also be interesting to have.)
@prabau $\mathbb{R}^\infty$ is the same as the subspace of finitely supported functions in $\mathbb{R}^\omega$. It fits the pattern of S000181, but I am not familiar with the term ' $\sigma$-product', and that page doesn't seem to have a reference with a general definition. What is your reference for $\sigma$-product?
Edit: Based on the definition of $\sigma$-product from the Encyclopedia of General Topology, on page 40, I believe you are correct.
@prabau Should I go ahead and switch things around so that the primary language is in terms of $\sigma$ products.
I'm not familiar with general properties of $\sigma$-products. Encyclopedia of General Topology cites the definition to Normality in Subsets of Product Spaces by Corson.
Encyclopedia of General Topology was also my source for this.
$\sigma$-product in also mentioned in @ccaruvana 's answer to https://math.stackexchange.com/questions/4736734/a-space-which-is-sigma-compact-but-neither-hemicompact-nor-second-countable, which was the basis for space S181.
I think it would make sense to use $\sigma$-product for the name of the space. It's a well known concept, and gives a short and unambiguous description, if one knows the terminology.
As for the specific notation, see the discussion Steven and I had for #899. As he mentions, there is no standardized notation here, so we picked one that seemed reasonable.
As for Munkres, it's a good book, but somewhat idiosyncratic in its notation and terminology sometimes. Was meant as an undergraduate text for MIT, not a graduate textbook. So we may not need to take his notation at face value.
Encyclopedia of General Topology was also my source for this.
σ -product in also mentioned in @ccaruvana 's answer to https://math.stackexchange.com/questions/4736734/a-space-which-is-sigma-compact-but-neither-hemicompact-nor-second-countable, which was the basis for space S181.
It's mentioned, but defined for a particular space. I think it makes sense to give references to general concepts which may not be known outside of general topology circles whenever they're used in special cases. Maybe it's worth clarifying what the level of knowledge expected from the ordinary user of pi-base is?
I was just mentioning the mathse post as an example using that terminology, but never suggested to include that. Referencing Encycl. of Gen Top is the right thing to do.
Maybe it's worth clarifying what the level of knowledge expected from the ordinary user of pi-base is?
I don't think any level of knowledge is expected. To verify properties of some examples, necessarily some advanced tools would have to be used, for others, elementary knowledge suffices.
In this particular example, lots of people in algebraic topology seem to use " $\mathbb{R}^\infty$ " as the name for this space. It'd be good to include multiple aliases.
@prabau I didn't think you did by the way. That was actually a comment about the page S000181
@Moniker1998 I like $\mathbb{R}^\infty$ as an alias for this reason, but I could be swayed either way.
@Moniker1998
In this particular example, lots of people in algebraic topology seem to use " R ∞ " as the name for this space.
Can you give an example of use in algebraic topology?
@prabau I didn't think you did by the way. That was actually a comment about the page S000181
@GeoffreySangston I agree that the page for S181 could mention the general construction of sigma-product with a suitable reference.
Can you give an example of use in algebraic topology?
Pay in mind that this is not my field, but this example is a colimit of $\mathbb{R}^n$ and a CW-complex, that's why its important in algebraic topology. See here for one example.
The superscript $\infty$ notation is common in bundle theory. See page 2 of Husemoller's book Fibre bundles, where $\mathbb{R}^\infty$, $\mathbb{C}^\infty$, $RP^\infty$, etc... are defined as direct limits (which he calls unions). It's also used in the book Characteristic Classes by Stasheff and Milnor; see page 62. It also appears in The Topology of Fiber Bundles Lecture Notes by Cohen (page 56 begins a list of direct limit spaces, and our space makes an appearance on page 59). I didn't actually see it in Steenrod's book on bundles in a quick skim through, however.
https://mathoverflow.net/questions/56363/list-of-classifying-spaces-and-covers
Also see Example 0.5. of Hatcher, on page 6, where $\mathbb{R}^\infty$ is defined.