[TOC]

0.1. 閉集合は全ての極限を含む

Let $A$ be a subset of a topological space $(X,\tau )$. Then $A$ is closed in $(X,\tau )$ iff $A$ contains all of its limit points.

証明 topology without tears > limit points

0.2. Def(連結:connected)

A topological space $(X,\tau )$ is said to be connected if the only clopen subsets of $X$ are $X$ and $\varnothing$.

It follows that $(X,\tau )$ is not connected (disconnected) if and only if there are non-empty open sets $A$ and $B$ such that $A\cap B=\varnothing$ and $A\cup B=X$.

0.3. Def(弧状連結:path-connected)

A topological space $(X,\tau )$ is said to be path-connected (or pathwise connected) if for each pair of (distinct) points $a$ and $b$ of $X$ there exists a continuous mapping $f:[0,1]\to (X,\tau )$, such that $f\left(0\right)=a$ and $f\left(1\right)=b$. The mapping $f$ is said to be a path joining $a$ to $b$

0.4. 弧状連結ならば連結

Every path-connected space is connected.

(proof)topology without tears > continuous mapping

0.5. Def(近傍:neighbourhood)

Let $(X,\tau )$ be a topological space, $N\subset X$, $p\in N$. Then $N$ is said to be a neighbourhood of a point $p$ if there exists an open set $U$ such that $p\in U\subset N$.

0.6. Def(孤立点:isolated_point)

Let $(X,\tau )$ be a topological space and $S\subset X$. Then a point $x\in S$ is called an isolated point of $S$ if there exists $U\in \tau$ such that $U\cap S=\left\{x\right\}$.

• This is equivalent to saying that the singleton $\left\{x\right\}$ is an open set in the topological space $S$ (considered as a subspace of $X$).(証明略)

0.7. Def(境界)

Let $(X,\tau )$ be any topological space and $A$ any subset of $X$. The largest open set contained in $A$ is called the interior of $A$ and is denoted by $\text{Int}\left(A\right)$ or $A^o$. The set $\text{Int}(X\setminus A)$, that is the interior of the complement of $A$, is denoted by $\text{Ext}\left(A\right)$, and is called the exterior of $A$. The set $\overline{A}\setminus A^o$ is called the boundary of $A$ and is denoted by $\partial A$.

0.8. 連結で空でない真部分集合は境界を持つ

Let $(X,\tau )$ be any connected topological space and $A$ any nonempty proper subset of $X$. Then the boundary of $A$, denoted by $\partial A$, is nonempty.

(proof)境界 $\partial A$が空集合だと仮定して矛盾を導く.$A$の閉包を$\overline{A}$, 内部を$A^o$と表すと境界の定義より$\overline{A}=A^o\cup \partial A$であるので左辺は閉集合,右辺は開集合となる.$X$は連結空間であるので開かつ閉集合は$X$もしくは$\varnothing$であるので$A\supset A^o=X$もしくは$A\subset \overline{A}=\varnothing$が成立するがこれは,$A$が空でないかつ真部分集合であることに矛盾.

0.9. ハウスドルフ空間上のコンパクト集合は閉集合

Let $(X,\tau )$ be a Hausdorff space and $K\subset X$ is compact. Then $K$ is closed.

Proof.

$x\in X\setminus K$を固定する.Hausdorff spaceであることから任意の$y\in K$に対して$U_y,V_y\in \tau$が存在して$x\in U_y,y\in V_y,U_y\cap V_y=\varnothing$を満たす.また,$\{V_y\mid y\in K\}$は$K$のopen coveringであるから$K$のコンパクト性よりfinite set $F$が存在して$\{V_y\mid y\in F\}$がfinite subcoverとなる.ここで,$U\equiv {\bigcap }_{y\in F}{U}_y$と定義すると$U$は$x$のneighborhoodであり$U\cap K=\varnothing$であるから$K$ はclosed.

0.10. 連続写像によるコンパクト空間の像はコンパクト

Let $(X,\tau )$ and $(X,{\tau }_1)$ be a continuous surjective map. If $(X,\tau )$ is compact, then $(Y,{\tau }_1)$ is compact.

(proof)topology without tears>compactness