Tootfinder

Totally paracompact spaces and the Menger covering property
Davide Giacopello, Maddalena Bonanzinga, Piotr Szewczak
https://arxiv.org/abs/2511.10252 https://arxiv.org/pdf/2511.10252 https://arxiv.org/html/2511.10252
arXiv:2511.10252v1 Announce Type: new
Abstract: A topological space is totally paracompact if any base of this space contains a locally finite subcover. We focus on a problem of Curtis whether in the class of regular Lindel\"of spaces total paracompactness is equivalent to the Menger covering property. To this end we consider topological spaces with certain dense subsets. It follows from our results that the above equivalence holds in the class of Lindel\"of GO-spaces defined on subsets of reals. We also provide a game-theoretical proof that any regular Menger space is totally paracompact and show that in the class of first-countable spaces the Menger game and a partial open neighborhood assignment game of Aurichi are equivalent. We also show that if $\mathfrak{b}=\omega_1$, then there is an uncountable subspace of the Sorgenfrey line whose all finite powers are Lindel\"of, which is a strengthening of a famous result due to Michael.
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Investment in subsea cable projects is expected to reach $13B between 2025 and 2027, almost 2x the amount invested between 2022 and 2024, driven by AI buildout (Magdalena Petrova/CNBC)
https://www.cnbc.com/2025/11/08/big-tech-ai-underwater-cables.html

Underwater cables are a vital piece of the AI buildout and internet — investment is booming
Over 95% of international data and voice traffic travels through subsea telecom cables. Tech giants Meta, Google, Amazon and Microsoft are investing for AI.

#caturday

Today, Vandenberg Spaceforce Base is best known for the booming SpaceX launches that send satellites into orbit.
But once upon a time, it was a hotbed for UFO sightings, quietly logged by the military.
This year, those sightings resurfaced.
Details of extraterrestrial encounters on the Central Coast were discussed in government hearings and explored in the newly released documentary
"The Age of Disclosure" (2025).
The film features government inside…

In New Doc ‘The Age of Disclosure,’ Military Servicemen Expose More Details About UFOs at Vandenberg
Depending on who you ask — and what you’re willing to believe — aliens are real. And they’ve visited Vandenberg Space Force Base in Lompoc more than once.

Urge Milan and Paris Fashion Weeks to Ban Fur and Wildlife Skins! #AnimalRights

Urge Milan and Paris Fashion Weeks to Ban Fur and Wildlife Skins!
Please urge organisers to act by banning cruel and environmentally destructive animal skins from its catwalks.

So the DC National Guard are an f-ing laughing stock of derelict traitorous fops. They couldn’t secure a lemonade stand and yet took and take the illegal order to deploy _against_ the American People.
11 f-ing shootings on the National Guard’s watch because the f-ing National Guard are not police and never should take a deployment to pretend to be.
You dishonorable scum. You will never live down the shame. Wastes of space.

11 people were shot at seven overnight incidents, police say
D.C. police are investigating the shootings of 11 people overnight at seven different locations. These shootings took place within nine hours.

The Crackdown on ICE Spotting Apps (with Joshua Aaron) https://www.404media.co/the-crackdown-on-ice-spotting-apps-with-joshua-aaron/