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Community Bot
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Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$. Does $D$ contain a non-trivial convergent sequence in $X$?

As shown by Ilya Bogdanov (Does every compact countable space contain a non-trivial convergent sequence?Does every compact countable space contain a non-trivial convergent sequence?) the answer is affirmative if $K$ is at most countable.

Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$. Does $D$ contain a non-trivial convergent sequence in $X$?

As shown by Ilya Bogdanov (Does every compact countable space contain a non-trivial convergent sequence?) the answer is affirmative if $K$ is at most countable.

Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$. Does $D$ contain a non-trivial convergent sequence in $X$?

As shown by Ilya Bogdanov (Does every compact countable space contain a non-trivial convergent sequence?) the answer is affirmative if $K$ is at most countable.

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Taras Banakh
Taras Banakh
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Convergent sequences in compact spaces

Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$. Does $D$ contain a non-trivial convergent sequence in $X$?

As shown by Ilya Bogdanov (Does every compact countable space contain a non-trivial convergent sequence?) the answer is affirmative if $K$ is at most countable.