Geodesic
Open, Connected Functions
Canadian mathematical bulletin, Tome 16 (1973) no. 1, pp. 57-60

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Recall that a function f:X→ Yis called connected if f(C) is connected for each connected subset C of X. These functions have been extensively studied. (See Sanderson [6].) A function f:X → Y is monotone if for each y ∊ Y, f-1(y) is connected. We shall use the techniques of multivalued functions to prove that if f: X→ Y is open and monotone onto Y, then f-1(C) is connected for each connected subset C of Y. This result is used to prove that the product of semilocally connected spaces is semilocally connected and that the image of a maximally connected space under an open, connected, monotone function is maximally connected. 
Friedler, Louis. Open, Connected Functions. Canadian mathematical bulletin, Tome 16 (1973) no. 1, pp. 57-60. doi: 10.4153/CMB-1973-012-9
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     title = {Open, {Connected} {Functions}},
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     doi = {10.4153/CMB-1973-012-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1973-012-9/}
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