Connectrdness concepts PDF

***[Connectrdness]*** Disconnected space , when (X,Ƭ) is topological space \ [*x* = *A* ∪ *B*]{.math.display}\ \ [*A* ≠ ⌀ , *B* ≠ ⌀]{.math.display}\ \ [*A*, *B* ∈ *τ*]{.math.display}\ \ [*A* ∩ *B* = ⌀]{.math.display}\ (X,Ƭ) is disconnected space , when (X,Ƭ) is topogical space [*X* ≠ *A* ∪ *...

***[Connectrdness]*** Disconnected space , when (X,Ƭ) is topological space \ [*x* = *A* ∪ *B*]{.math.display}\ \ [*A* ≠ ⌀ , *B* ≠ ⌀]{.math.display}\ \ [*A*, *B* ∈ *τ*]{.math.display}\ \ [*A* ∩ *B* = ⌀]{.math.display}\ (X,Ƭ) is disconnected space , when (X,Ƭ) is topogical space [*X* ≠ *A* ∪ *B*]{.math.inline} \ [*A*, *B* ∈ *Ƭ*]{.math.display}\ \ [*A* ∩ *B* ≠ 0]{.math.display}\ \ [*A* ≠ ⌀ = *B*]{.math.display}\ **Example** [*x* = {1,2,3}]{.math.inline} Ƭ= { x,Ф,{1} } is (x,Ƭ) connected X=X [ ∪ {1}             *But* *x* ∩ {1} ≠ ⌀]{.math.inline} (x,Ƭ) is connected space If [*A*, *A*^*C*^ ∈ *τ* → (*x*,*τ*) is disconnected]{.math.inline} [*A* ∪ *A*^*C*^ = *x*]{.math.inline} \ [*A* ∩ *A*^*C*^ = ⌀]{.math.display}\ **[Remarks]** : (x,D) → disconnected space , D= discrete topology ([*x*, *τ*) *is* *topological* *apace*             *τ* = *ʄ* ]{.math.inline} Open set =close set [*A*, *A*^*C*^ ∈ *τ*]{.math.inline} [(*x*,*τ*)disconnected]{.math.inline} ([*x*, *I*)            *x* = *x* ∪ ⌀]{.math.inline} , I = inscrete topology [*β* = ⌀              ∴ *τ* *is* *connected*]{.math.inline} **Example ։**Let [*τ* = {*x*,⌀,{1},{2,3}}]{.math.inline} ([*x*, *τ*) *topo* *space* :]{.math.inline} Is Ƭ connected or disconnected A[ ∪ *B* = {1} ∪ {2,3} = {1,2,3}]{.math.inline} (X,Ƭ) is disconnected space [REMARKS ] *Let T be clopen subset of R , then either T=R or T= Ø.* *Let (X, τ ) be a topological space. Then it is said to be connected if the only clopen subsets of X are X and Ø.* *The topological space R is connected.*