We give a pointwise version of sensitivity in terms of open covers for a semiflow ( T ,  X ) of a topological semigroup T on a Hausdorff space X and call it a Hausdorff sensitive point. If ( X , U ) is a uniform space with topology τ , then the definition of Hausdorff sensitivity for ( T , ( X , τ ) ) gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow ( T ,  X ) on a compact Hausdorff space X , these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are T -invariant if T is a C -semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow ( T ,  X ) on a topological space X and show that if ( T ,  X ) is a topologically equicontinuous pair in ( x ,  y ), for all y ∈ X , then Tx ¯ = D T ( x ) where D T ( x ) = ⋂ { TU ¯ : for all open neighborhoods U of x } . We prove for a topologically transitive semiflow ( T ,  X ) of a C -semigroup T on a regular space X with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of C -semigroup T on a regular space X with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if X is a regular space and ( T ,  X ) is not a topologically equicontinuous pair in ( x ,  y ), then x is a Hausdorff sensitive point for ( T ,  X ). Hence, a minimal semiflow of a C -semigroup T on a regular space X is either topologically equicontinuous or topologically sensitive.