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Components and Partitions1. PartitionsLet $X$ be a space. A partition of $X$ is the collection of disjoint subspaces of $X$ such that the union equals $X$. Example: $[0,1]$ and $(1,2]$ are partitions of $[0,2]$.Example: $[0,1]$ and $[1,2]$ are not partitions because they are not disjoint.2. ComponentsComponents are equivalence classes by defining the equivalence relation $\sim$ so that $x\sim y$..

위상수학 공부하다가 bounded의 정의가 모호하다는 것을 발견했다.인터넷에서 흥미로운 글을 발견했다. 참고: https://math.stackexchange.com/questions/1500897/bounded-set-in-metric-vs-ordered-spaces The term "bounded" is overloaded. 여기서 overloaded는 과적했다는 뜻인데, 컴퓨터 공부한 사람들은 한 함수가 여러 가지 방법으로 정의되었다는 걸 뜻한다고 알고 있을 거다.이 문장에서 overloaded 역시 그런 뜻인데 용어 하나에 여러가지 뜻이 담겨 있고 문맥에 따라 어느 뜻으로 쓰였는지 판단할 수 있다는 거다. 다음은 bounded이라는 용어에 대한 나의 설명.1. In METRIC TOPOLOGY, ..

Munkres 54.8 Let $p:E\rightarrow B$ be a covering map. If $B$ is simply connected and $E$ is path connected, then $p$ is a homeomorphism.Proof.Let $x_1, x_2\in E$ and $\tilde{f}$ be a path from $x_1$ to $x_2$. Then $f=p\circ\tilde{f}$ is a path from $p(x_1)$ to $p(x_2)$ and by the path lifting theorem, $\tilde{f}$ is the unique path lifting of $f$. Now, assume $p(x_1)=p(x_2)$, then $f$ is a loop..

Munkres 53.6(b)Let $p:E\rightarrow B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite for all $b\in B$, then $E$ is compact.Proof.Let $\{U_\alpha\}$ be an open cover of $E$. *Recall: a covering map is an open map.If $E$ is an $n$-fold covering of $B$, $p^{-1}(b)=\{e_1, e_2,...,e_n\}$, there exists a finite subset of $\{U_\alpha\}$ that covers $p^{-1}(b)$, say $\{U_1, U_2, ...,U_n\..

Let $U\subset B$ be evenly covered by $p:E\rightarrow B$ and $\{V_\alpha\}$ be the partition of $p^{-1}(U)$ into pieces. Number of elements of $\{V_\alpha\}$ is equal to the number of elements of $p^{-1}(b)$ for all $b\in U$.Because each of $p|_{V_\alpha}$ is a homeomorphism, exactly one element of $p(b)$ exists in $V_\alpha$. No more, no less.

Proposition. Let $p:E\rightarrow B$ a covering map. If $U\subset B$ is evenly covered by p and $U'\subset U$ is open, $U'$ is evenly covered by p.Proof.Let $\{V_\alpha\}$ be the partition of $p^{-1}(U)$ into pieces and $p|_{V_\alpha}:V_\alpha\rightarrow U$, a homeomorphism. Let $V_\alpha=(p|_{V_\alpha})^{-1}(U)$ are disjoint, so $(p|_{V_\alpha})^{-1}(U')$ are disjoint and $\bigcup (p|_{V_\alpha..

Proposition.Let $x_0$ and $x_1$ be points of the path-connected space $X$. Show that $\pi_1(X,x_0)$ is abelian if and only if for every pair of $a$ and $b$ of paths from $x_0$ to $x_1$, we have $\hat{a}=\hat{b}$.Proof.Let $h=f*a$.First, start with $\hat{a}=\hat{b}$. Then $\hat{a}([f]*[g])=\hat{h}([f]*[g])=[\bar{h}]*[f]*[g]*[h]=[\bar{a}]*[\bar{f}]*[f]*[g]*[f]*[a]$$=[\bar{a}]*[g]*[f]*[a]=\hat{a}(..

PropositionLet $p:E\rightarrow B$ be continuous and surjective. Suppose that $U$ is an open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ into pieces is unique.Proof.Let $x_0\in p^{-1}(U)$, and $\{V_{\alpha}\}$ and $\{W_{\alpha}\}$ be two possible partition of $p^{-1}(U)$ into pieces. Then $x_0\in V_0$ and $x_0\in W_0$ for some $V_0$ a..

1편: https://wangdo.tistory.com/62 Limit point 이해하기Limit(극한)의 정의를 살펴봅시다.고등학교 수준에서는 너무 깊이 들어가면 복잡하기만 하니 당연히 간편하게 ~~에 가까워진다 정도로만 배웠겠지만 (엡실론 델타 논법도 고등학교 수준에선 배우wangdo.tistory.com Limit point 이해하기 시리즈 2입니다. 1편에서는 limit point를 극한이 존재할 수 있는 점이라고 설명했었다. 다른 관점에서 보면 극한값 그 자체라고 볼 수도 있겠다.{1/n | n은 자연수}라는 집합을 보아라.이 집합의 limit point는 0 하나이다.그리고 수열 aₙ=1/n이 어디로 수렴하는지 보아라.0이다! Standard topology 상에서 이야기하자면어떤 집합의 서..

TheoremLet f : X → Y; let Y be compact Hausdorff. f is continuous if and only if the graph of f, Gf = { x × f(x) | x ∈ X } is closed in X × Y. Proof Start with f is continuous. (the if only part)Fix a point x ∈ X. Let y be a point in Y such that y₀ ≠ f(x) (so that x × y ∉ Gf)Let Nf be a neighborhood of f(x) and Ny be a neighborhood of y in Y such that Nf ∩ Ny = Ø. (They exist because Y is Hausdo..