In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $ k $-bounded sober spaces and weakly sober spaces. For $ T_{0} $ spaces $ X $ and $ Y $, it is proved that $ Y $ is bounded sober iff the function space $ {\bf{Top}}(X, Y) $ of all continuous functions $ f : X\longrightarrow Y $ equipped with the pointwise convergence topology is bounded sober iff $ {\bf{Top}}(X, Y) $ equipped with the Isbell topology is bounded sober. But for a $ k $-bounded sober space $ X $, the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology may not be $ k $-bounded sober. It is shown that if the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $ Y $ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.
Citation: Xiaoyuan Zhang, Meng Bao, Xiaoquan Xu. On function spaces related to some kinds of weakly sober spaces[J]. AIMS Mathematics, 2022, 7(5): 9311-9324. doi: 10.3934/math.2022516
In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $ k $-bounded sober spaces and weakly sober spaces. For $ T_{0} $ spaces $ X $ and $ Y $, it is proved that $ Y $ is bounded sober iff the function space $ {\bf{Top}}(X, Y) $ of all continuous functions $ f : X\longrightarrow Y $ equipped with the pointwise convergence topology is bounded sober iff $ {\bf{Top}}(X, Y) $ equipped with the Isbell topology is bounded sober. But for a $ k $-bounded sober space $ X $, the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology may not be $ k $-bounded sober. It is shown that if the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $ Y $ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.
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Xiaoyuan Zhang, Meng Bao, Xiaoquan Xu. On function spaces related to some kinds of weakly sober spaces[J]. AIMS Mathematics, 2022, 7(5): 9311-9324. doi: 10.3934/math.2022516
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