One-point compactification and Discourse on Method

近幾天看到兩項讓我在辦公室內微微驚呼出聲的東西。首先是拓樸上的 one-point compactification,本來應該上星期就寫,但一直被別的主題擠掉。One-point compactification 顧名思義,一個只是 locally compact 的 Hausdorff space 只要補上一個點然後簡單地拓樸化,這個新的 space 就變成 compact 了!不僅如此,所有只多一個點的 compactification 全部都是 homeomorphic,所以 one-point compactification 是 unique up to homeomorphism!這麼單純卻又有這麼好的性質,可惜當時旁邊沒有人可以分享我的驚喜 XD。相較之下,號稱點集拓樸第一個 non-trivial 的命題 Urysohn's lemma 以及其後的 Urysohn's metrisation theorem 就真的是 non-trivial,我因此還看不出那個 metric 到底是怎麼變出來的。

另外也讀完 Descartes 的《Discourse on Method》第一部份。這根本就是 Descartes 的學思歷程嘛!一路讀下來,我覺得我的思路實在和 Descartes 很接近,而且根源應該不是因為讀 Meditations(雖然讀 Meditations 對往後的發展一定有影響)。看到 Part One 最後,Descartes 說

[...] I gradually extricated myself from many errors powerful enough to darken our natural intelligence, and incapacitate us in great measure from listening to reason. But after I had been occupied several years in thus studying the book of the world, and in essaying to gather some experience, I at length resolved to make myself an object of study, and to employ all the powers of my mind in choosing the paths I ought to follow [...]
我看到這最後一段不微微驚呼出聲也不行了。還是引 Descartes 的話:
I was aware [...] that the perusal of all excellent books is, as it were, to interview with the noblest men of past ages, who have written them, and even a studied interview, in which are discovered to us only their choicest thoughts[.]

不過 Descartes 要講的重點是讀太多別人的東西不好 XD。

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