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connected space compact space

Conneted space

If there are disjoint open subsets \(U \cup V = X\). The space X is disconnected. Otherwise connected.

Path connected space

Every elements \(x,y \in X\) have image of \(f:[a,b] \rightarrow X\) that \(f(a) = x, f(b)=y\) is in \(X\).

Component

Relation class \([x]\) is a subset of of \(X\) those elements is also elements of connected subspaces of containing \(x\). Connected space has only one component. \(\mathbb{R}\) except \({0}\) is disjoint space and has two componets \({(-\infty,0)}\) and \({(0, \infty)}\).

Path component

Compact space

Every open cover has at least one finite open subcovers. \([0,a)\) is open in subspace \([0,1]\) of \(\mathbb{R}\). Subset \(A \subset X\) topology \(\tau'\) is \(\tau \cap A\).