## 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 \(\tau \cap A\).