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ABSTRACT
When discussing the concept of connectedness, we often come across the equivalent
criterion that a space is connected if and only if any continuous map from it to the
discrete space {0,1} is constant. It would be interesting to see what concept arises if
the discrete space of two points is replaced by some other spaces. Let Z be a T1
topological space having more than one point, then a space X is said to be Zconnected if and only if any continuous map from X to Z is constant. It can be shown
that this idea generates some stronger notion of connectedness and this stronger
notion has many similarities with the usual connectedness. The first nontrivial
example of Z-connected space can be constructed by taking Z to be the space ℒ of
integers equipped with the complement finite topology. Define a space X to be
strongly connected if and only if it is ℒ–connected. This strongly connected space has
many interesting properties.