Connected Space

Definition

A Topological Space $(X,\mathcal{T})$ is connected if it can not be represented as the union of two disjoint, non-empty, open or closed subsets. $\nexists A,B\in \mathcal{T}\text{s.t.}A,B\ne \varnothing \text{and}A\uplus B=X⟺\nexists (X\setminus A),(X\setminus B)\in \mathcal{T}\text{s.t.}A,B\ne \varnothing \text{and}A\uplus B=X$ where $\uplus$ is a disjoint union.

Or, equivalently, if $\varnothing ,X$ are the only open and closed subsets.

Examples

Topologist’s sine curve

Consider a Subspace Topology $(X,{\mathcal{T}}_X)$, where $X=\{(x,\sin (\frac{1}{x}))|0<x\le 1\}\cup (0\times [-1,1])$, of a Standard Topology $({\mathbb{R}}^2,\mathcal{U})$ on real plane. The Topological Space $(X,{\mathcal{T}}_X)$ is connected, but not path connected.

Consider a set $\{a,b\}$.

• The Topological Space with discrete topology $(X,\{\varnothing ,a,b,X\})$ is not connected.
• The Topological Space with trivial topology $(X,\{\varnothing X\})$ is connected and path connected by a path $f=\{\begin{array}{ll}a& [0,1)\\ b& 1\end{array}$.

Consider a Subspace Topology $(\mathbb{Q},{\mathcal{T}}^{\prime })$ of a Standard Topology $(\mathbb{R},\mathcal{U})$ on real number. The Topological Space $(\mathbb{Q},{\mathcal{T}}^{\prime })$ is not connected by a separation $\{(\mathbb{Q}\cap (-\infty ,\sqrt{2})),(\mathbb{Q}\cap (\sqrt{2},\infty ))\}$

Consider a Subspace Topology $(X,{\mathcal{T}}_X)$, where $X=\{(x,y)|y=0\}\cup \{(x,\frac{1}{x}|x>0\}$, of a Standard Topology $({\mathbb{R}}^2,\mathcal{U})$ on real plane. The Topological Space $(X,{\mathcal{T}}_X)$ is not connected by a separation $\{U,V\}$.

Facts

Every path connected space is a Connected Space.

Link to original

Consider a Subspace Topology $(Y,{\mathcal{T}}_Y)$ of a Topological Space $(X,\mathcal{T})$. Then, $A,B\in {\mathcal{T}}_Y\text{s.t.}A,B\ne \varnothing \text{and}A\uplus B=Y$ is a separation of $Y$ if and only if $A\uplus B=Y\text{and}{A}^{\prime }\cap B=A\cap B^{\prime }=\varnothing$. where $\uplus$ is a disjoint union, $A^{\prime }$ is the set of all limit points of $A$.

Consider a Subspace Topology $(Y,{\mathcal{T}}_Y)$ of a Topological Space $(X,\mathcal{T})$ and a separation $\{C,D\}$ of $X$. If $(Y,{\mathcal{T}}_Y)$ is a connected subspace, then $Y\subset C\text{or}Y\subset D$.

Consider a collection of connected subspaces $\{U_i{\}}_{i\in I}\text{s.t.}{U}_i\subset X$ of a Topological Space $(X,\mathcal{T})$. If the subspaces have a point in common $\exists x\in X,\text{s.t.}x\in \bigcap _{i\in I}{U}_i\iff \bigcap _{i\in I}{U}_i\ne \varnothing$, then the union of the collection $\bigcup _{i\in I}{U}_i$ is connected.

Consider a collection of connected subspaces $\{U_i{\}}_{i\in I}\text{s.t.}{U}_i\subset X$ of a Topological Space $(X,\mathcal{T})$, and a connected subspace $B$. If $\forall i\in I,U_i\cap B\ne \varnothing$, then $B\cup \left(\bigcup _{i\in I}{U}_i\right)$ is connected.

Consider a connected Subspace Topology $(A,{\mathcal{T}}_A)$ of a Topological Space $(X,\mathcal{T})$. For some Subspace Topology $B$, if $A\subseteq B\subseteq A^{\prime }$, then $B$ is connected.

Consider two topological spaces $(X,{\mathcal{T}}_X),(Y,{\mathcal{T}}_Y)$. If $(X,{\mathcal{T}}_X)$ is connected and there exists a Continuous Function $f:X\to Y$, then $Y$ is a connected space.

Consider a continuous function $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces. If $(X,{\mathcal{T}}_X)$ is connected, then $f(X)$ is a connected subspace in $(Y,{\mathcal{T}}_Y)$.

Consider a Topological Space $(X,\mathcal{T})$ and a subspace $Y\subset X$. If $Y$ is connected, then its Closure $\bar{Y}$ is also connected.

The Product Space of any collection of connected spaces is a connected space.