General topology | Properties of topological spaces

First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods. (Wikipedia).

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This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.

From playlist Proofs

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From playlist Column and Null Space

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This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.

From playlist Proofs

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What is a Manifold? Lesson 4: Countability and Continuity

In this lesson we review the idea of first and second countability. Also, we study the topological definition of a continuous function and then define a homeomorphism.

From playlist What is a Manifold?

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1.11.3 Countable Sets: Video

MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015

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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

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PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis

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From playlist Manifolds

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Manifolds - Part 6 - Second-Countable Space [dark version]

Support the channel on Steady: https://steadyhq.com/en/brightsideofmaths Or support me via PayPal: https://paypal.me/brightmaths Or via Ko-fi: https://ko-fi.com/thebrightsideofmathematics Or via Patreon: https://www.patreon.com/bsom Or via other methods: https://thebrightsideofmathematics.

From playlist Manifolds [dark version]

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This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.

From playlist Proofs

Related pages

Topological space | Metric space | Closure (topology) | Topology | Order topology | Limit of a function | Quotient space (topology) | Axiom of countability | Second-countable space | Sequence | Sequentially compact space | Countably compact space | Mathematics | First uncountable ordinal | Fréchet–Urysohn space | Without loss of generality | Subset | Limit of a sequence | Sequential space | Compactly generated space | Discrete space | Neighbourhood system