# 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 $${(0, \infty)}$$.

## 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 $$\tau \cap A$$.