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I came up with some conditions for dense subsets of topological sets and I assume the conditions have names already but I have no idea what they are.

Suppose I have a topological set $X$ and a subset $A \subset X$ that is dense in $X$. I am looking for names of the following conditions on $A$.

To simplify things, let's just say $X$ is a metric space with distance $d(\cdot, \cdot )$

In all definitions suppose there is a point $x \in X$ that is fixed:

Condition C1: For any $\varepsilon > 0$, there exists a known procedure to find an element $a \in A$ such that $\varepsilon > d(a, x)$.

Condition C2: For any $\varepsilon > 0$, we know there exists a procedure to find an element $a \in A$ such that $\varepsilon > d(a, x)$, but we don't know what the procedure is.

Condition C3: There provably does not exist any single procedure that will identify a member of $a \in A$ such that $\varepsilon > d(a, x)$ for all $\varepsilon > 0$.

Do conditions C1, C2, and C3 have names?

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    $\begingroup$ Presumably every time you wrote $d(a, x)$ you meant $d(a, x) < \varepsilon$. The first term that comes to mind is "computably dense" (for the condition which is either C1 or C2) which has google search results: google.com/search?q=%22computably+dense%22 $\endgroup$ Commented Mar 24 at 5:12
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    $\begingroup$ @QiaochuYuan Thank you! Oddly enough, this post is now the top result for "Computably dense". I wonder if C3 has been studied. $\endgroup$ Commented Mar 24 at 10:54
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    $\begingroup$ What you're describing reminds me of some concerns I've had with Sierpiński's use of the word "effective". See my answer to A question about a proof in one of Sierpiński's papers, especially the paragraph containing the sentence "However, difficulties seem to arise on closer inspection.". $\endgroup$ Commented Apr 8 at 1:56

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