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We consider a topological space βŸ¨π‘‹, 𝜏 (β„±)⟩, where 𝑋 = {𝑝 βˆ—} βˆͺ [πœ” Γ…~ πœ”] and β„± βŠ† πœ”πœ”. Each point in πœ” Γ…~ πœ” is isolated and a neighborhood of π‘βˆ— has the form {π‘βˆ—}βˆͺ{βŸ¨π‘–, π‘—βŸ© : 𝑖 β‰₯ 𝑛, 𝑗 β‰₯ 𝑓(𝑖)} for some 𝑛 ∈ πœ” and 𝑓 ∈ β„±. We show that there are subsets β„± and 𝒒 of πœ”πœ” such that β„± is not bounded, 𝒒 is bounded, yet βŸ¨π‘‹, 𝜏 (β„±)⟩ and βŸ¨π‘‹, 𝜏 (𝒒)⟩ are homeomorphic. This answers a question of the second author posed in A space topologized by functions from πœ” to πœ”, [Topology Proc. 34 (2009), 161–166].