Descriptive set theory | General topology

Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory. Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval (0, 1) is Polish. Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable Polish space has the cardinality of the continuum. Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces. (Wikipedia).

Video thumbnail

What is space?

What exactly is space? Brian Greene explains what the "stuff" around us is. Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https:

From playlist Science Unplugged: Physics

Video thumbnail

Gdansk UFO July 4 2008

Gdansk UFO July 4 2008 - flight in town

From playlist Projects & Installations

Video thumbnail

EARTH FROM SPACE: Like You've Never Seen Before

Showing cities all over the world from orbit from North America to India with a dramatic Beethoven soundtrack.

From playlist Space Videos

Video thumbnail

Searching for Hidden Nazi Treasure in Poland | What On Earth | Science Channel

Satellite images capture a series of patterns that may lead to the Nazi's hidden treasure on Poland territory. #sciencechannel #whatonearth Stream What On Earth: https://www.discoveryplus.com/show/what-on-earth About What On Earth: Hundreds of satellites orbit Earth and capture the wo

From playlist Recent Uploads

Video thumbnail

What is spacetime?

"Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https://twitter.com/worldscienceu"

From playlist Science Unplugged: Special Relativity

Video thumbnail

Interstellar flight: 10 Hard Facts

We can build powerful rockets able to carry people and machines into orbit, or even vault them to the moon. But our fastest spacecraft don't hold a candle to the distances that define Interstellar Flight. So what's on the drawing boards? What futuristic designs and fuel options promise to

From playlist Spaceten

Video thumbnail

NuSTAR in Space

NASA's newest X-ray telescope will have a lengthy structure that unfolds in space, allowing it to see high-energy objects like feeding black holes.

From playlist NuSTAR

Video thumbnail

The Space Elevator

Alan Chan's Masterpiece about Space Exploration and the Space Elevator

From playlist The Best of Futurism on YouTube

Video thumbnail

Alex SIMPSON - Probability sheaves

In [2], Tao observes that the probability theory concerns itself with properties that are \preserved with respect to extension of the underlying sample space", in much the same way that modern geometry concerns itself with properties that are invariant with respect to underlying symmetries

From playlist Topos à l'IHES

Video thumbnail

Joscha Prochno: The large deviations approach to high-dimensional convex bodies, Lecture I

Given any isotropic convex body in high dimension, it is known that its typical random projections will be approximately standard Gaussian. The universality in this central limit perspective restricts the information that can be retrieved from the lower-dimensional projections. In contrast

From playlist Workshop: High dimensional spatial random systems

Video thumbnail

Luca Motto Ros: Towards the « right » generalization of descriptive set theory to...

Recording during the meeting "15th International Luminy Workshop in Set Theory" the September 26, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's A

From playlist Logic and Foundations

Video thumbnail

The Most Advanced Mirror in the World

Welcome to the most advanced mirror in the world. So, what does it do? In this episode of SciShow, Hank describes the five reasons why infrared telescopes were so difficult to build, and how scientists finally developed the legendary James Webb Space Telescope. Join us! Many thanks to S

From playlist Uploads

Video thumbnail

Warszawa, 1978 rok - Uroczyste powitanie kosmonautów: Hermaszewskiego i Klimuka [1/2]

Uroczystości w lipcu 1978 roku w Warszawie związane z powitaniem kosmonautów: Mirosława Hermaszewskiego oraz Piotra Klimuka po locie w kosmos rakiety Sojuz-30. Zobaczymy przejazd przez Warszawę, uroczyste odznaczenie z rąk Władysława Gomułki oraz Edwarda Gierka w stolicy czy wreszcie tłum

From playlist Warszawa - stare filmy

Video thumbnail

Battle of Warsaw - Turning Point of Polish-Soviet War I THE GREAT WAR 1920

Sign up for Curiosity Stream and get Nebula bundled in: https://curiositystream.com/thegreatwar In the summer of 1920 the new Poland under Josef Pilsduksi stood with their backs to Warsaw against the Red Army. The Bolsheviks had advanced in the North and in the South and some of the Sovie

From playlist Russian Civil War(s) 1917 - 1923

Video thumbnail

Adam Savage's Favorite Tools: Motorized Angle Polisher

As seen in his recent restoration of his Nagra IV-S plastic cover, Adam demonstrates the usefulness of a motorized angle polisher like the one made by Proxxon in his buffing of a spacesuit helmet! The variable speed of this kind of handheld polisher works great with Novus multi-stage plast

From playlist Adam Savage's Favorite Tools

Video thumbnail

Norbert Müller : Wrapping in exact real arithmetic

Abstract : A serious problem common to all interval algorithms is that they suffer from wrapping effects, i.e. unnecessary growth of approximations during a computation. This is essentially connected to functional dependencies inside vectors of data computed from the same inputs. Reducing

From playlist SPECIAL 7th European congress of Mathematics Berlin 2016.

Video thumbnail

Gdansk UFO - test flight

On-board camera footage from the Gdansk UFO test-flight. Rather shaky due to generator vibration. This is the take-off & landing location : http://tinyurl.com/56lmqz

From playlist Projects & Installations

Video thumbnail

Eli Glasner : A universal hypercyclic representation

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Dynamical Systems and Ordinary Differential Equations

Video thumbnail

Space Power Stations, Robots, Space Life Structures: Future of Russian Space

Future of Russian Space Program: Reflectors that light up Siberia, Solar Power Stations in orbit, Robots to help build large space structures and the International Space Station. An interview with Vladislav Rutkovsky - pioneer in Soviet space program and repected Control Engineer. Russia

From playlist Russian Engineering

Video thumbnail

Why is this Space Telescope so Tiny?

Optical Engineer Rik ter Horst shows us how he makes very small telescopes (at home) which are intended for use in micro-satellites. Contents: 0:00 Intro 1:06 About telescopes and focal length 3:35 The Cassegrain telescope 4:38 The Schmidt-Cassegrain telescope 5:18 The monolithic telesco

From playlist optics

Related pages

Cantor space | Topological space | Metric space | Lie group | Hilbert cube | Borel isomorphism | Separable space | Inner regular measure | Strong operator topology | Forgetful functor | Banach space | Cardinality of the continuum | Open mapping theorem (functional analysis) | Hans Freudenthal | Perfect set | Hausdorff space | Disjoint union | Completely metrizable space | Standard Borel space | Descriptive set theory | Determinacy | Alfred Tarski | Stefan Banach | Dense set | General topology | Locally finite measure | Probability measure | Baire space (set theory) | Polish space | Bijection | Property of Baire | Borel equivalence relation | Hilbert space | Radon measure | Irrational number | Probability theory | Subspace topology | Closed graph theorem | Choquet game | Johann Radon | Borel measure | Suslin operation