[數撚圍爐區] 重口味的東西真的是重口味嗎? (109)

都係D咩close enough, there is an open nhd, blah blah blah
d epsilon delta無哂架啦 除非你做hard analysis先會見到

有我覺得感覺上operational多啲 可以較大細 而唔係存在nhd
已證T1 <=> {x} is closed

Coarse(weak), fine(strong) topology
coarsest topology contain S (subset of P(X))can be defined in two way, one by intersection, one by operation, find all finite intersections of S first, and arbitrarily union of them after, then we can generate all possible open set of S operationally. We can prove it is really a topology, as 3 axiom already done in the procedure! And these two definitions are actually equivalent, though one may think the former is quite arbitrary, the latter is kind of doable
Coarsest topology contain S is at the same time topology of X (i.e. not proper subset), we call S subbasis wrt to it
If we dont need finite intersections and just union, and we get the topology of X, we then call it basis. For example, we can make use of the recipe above, and call all finite intersections of S = M, and if we get the topology of X, then M is actually basis
Here, we can determine all element of our intended subset I of P(X) as open, and by calling I open, we generate a topology

之後到local basis

聽日應該係first/second countability

Compact space is sub class of Lindelof space?
Separable - Arzela Ascoli thm? example: R有Q
第1 countable
對於所有點同任意冚住一點嘅nhd U，我地都有采自countable嘅local basis嘅V喺U入面（我諗唔到有咩唔係，唔係好自然咩）

第2 countable，呢個X嘅topology可以用countable basis generate返出嚟

第2推第1 obvious
第2推separable，所有basis都立一個x，叫呢啲x做countable dense subset，是但一個open set都係用basis做出嚟，咁non empty嘅open set點都撞到個x

第2推lindelof，如果X可以open cover，每個open set本質上都只係countable basis generate出嚟。咁其實即係話我地兵在精不在多，用番有齊呢啲countable basis嘅open cover就夠啦，i.e.我地搵有呢個Bi (i.e. reindex version as it is countable)嘅open cover Ui, union呢啲Ui，就等同於union我地同一個building block，咁就既cover X，又係countably many

Categorical logic

Initial topology -> subspace and product topology
由gi is continuous for all i 推出 g is continuous is by universal property
由g is continuous 推出gi is continuous for all i is by composition of continuous function is continuous
咁我就可以話你聽點解R to R^n is continuous = each component being continuous

好重口

邊度摷到

Tmr possibly final topo and example
Btw感覺point set topo要好set theory就要好，好彩之前打咗底

感覺唔到會有engineering application
即係classical CS halting problem咁 知左對實際engineering冇用

主要都係探索boundary of computing

Final topo, quotient topo, quotient mapping, R-saturated
We can do in that way, let f: X -> Y is continuous and
g: X -> X/[f] is surjective while X is equipped with final topology(quotient topology), then g is quotient mapping
Hence, we can prove there exist unique continuous mapping h: X/[f] -> Y by some simple proof
If you are not lucky today and run into a situation which the equivalence relation is not f, then its well-definedness required f is equal if g is equal
Example: i need to find one in book

Abstract algebra嘅deja vu

我的故事不同：
我小學的時候極度討厭數學，成績徘徊合格同唔合格邊緣。
但到左中學，有一位好好的老師，啟發左我對數學的興趣，都俾左信心我，例如我知道數學對科學咁緊要，都可以解釋到日常生活的問題，自此突飛猛進。後尾靠佢攞A入醫學院。而家諗番，小學的唔係數學，係算術，好technical,但中學的先係真正數學，係Art,我好鍾意

Googology

川普砍UCLA經費「數學界莫扎特」不滿、放話離開美國

世界最著名數學家之一、有「數學界的莫扎特」美譽的洛杉磯加州大學(UCLA)學者陶哲軒(Terence Tao)平時不談政治，但7月聯邦扣下給洛杉磯加大的5億8400萬元補助款後，陶哲軒開口抨擊川普政府，稱聯邦大砍學術經費，已對美國學術研究造成「生存威脅」，趨勢不改，包括他等科學家都會離美。

澳洲出生的陶哲軒表示，他發現這屆川普政府對改變科學界生態特別激進，比川普第一次當總統還過火，這並不尋常，而且很多人並不了解那麼做正造成的傷害。

陶哲軒是目前公開反對聯邦大砍學術經費的最知名數學家及學術人士之一，他表示川普政府的做法已對他的學域及更廣的學術，造成生存威脅；他認為目前公開倡導，反對川普政府做法，比做研究更為重要。

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