Monico (2024) · Lemma 3.2

# Point-Set Separator

Watch how \(\mathcal{N}_2\) separates a point \(\mathbf{x}_0\) from a closed set \(B\) using compactness and finite covers.

Lemma 3.2: Let \(B \subset K\) be closed and \(\mathbf{x}_0 \in K \setminus B\). For each \(\varepsilon > 0\), there exists \(g \in \mathcal{N}_2\) such that \(g > 1 - \varepsilon\) on \(B\) and \(g(\mathbf{x}_0) < \varepsilon\). Drag \(\mathbf{x}_0\) and step through the construction.
\(g(\mathbf{x}_0)\)
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\(\min g\) on \(B\)
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Cover size \(N\)
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Neurons
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Preset shapes for B
Parameters
ε = 0.10
Cover points N = 12
Construction step
Explanation
Result The heatmap shows \(g(\mathbf{x}) = \sum_{j=1}^N F_j(\mathbf{x})\). Bright = high value. Point \(\mathbf{x}_0\) should be dark (below \(\varepsilon\)), set \(B\) should be bright (above \(1-\varepsilon\)).