Closeness follows density

• Six cities, three patterns

  Málaga is where proximity most favors lower incomes: the lowest-income quintile has an index of 48.4 and the highest, 35.4 (−13.0; 90% interval, −18.6 to −7.0). In Madrid too, by a smaller margin (−4.9). Seville leans toward higher incomes (+5.9, at the edge of zero). Barcelona (+0.2), Valencia (+4.1) and Zaragoza (0.0) show no clear relationship.

• Closeness follows density

  In all six cities, denser sections have more proximity (rank correlation of 0.17 to 0.70) and higher-income sections are less dense (−0.41 to −0.21). Once density is accounted for, the relationship between income and proximity becomes positive in all six (0.06 to 0.28): at equal density, higher incomes have slightly more nearby.

• In Seville, lower incomes have less of everything nearby

  All six functions within fifteen minutes reach 55.7% of the lowest-income quintile and 77.4% of the highest, the largest gap of the six cities. 21.1% of its population lives with low proximity and low income, in neighborhoods such as Polígono Sur, Parque Alcosa, Los Pájaros and Palmete. Without culture and health, the index gap falls to +1.7.

• Public facilities, closer to lower incomes; health, to higher ones

  Culture is closer to lower incomes in Madrid, Barcelona, Valencia and Málaga; education, in Madrid, Zaragoza and Málaga. Health is closer to higher incomes in all six (+2.3 to +22.3 percentile points): in OpenStreetMap it includes private practices and clinics, which concentrate where incomes are higher.

• Barcelona does not distinguish by income

  60.1% of its population is in the third with the most proximity across the six cities, whatever their income; between the extreme quintiles there are +0.2 points.

• Low proximity and low income

  845,017 people in the six cities live in sections with low proximity and low income: 21.1% of Seville, 13.3% of Valencia and 11.4% of Madrid. This is the case that calls for attention first.

Start with the sections where low proximity and low income coincide, above all in Seville (Polígono Sur, Parque Alcosa and Los Pájaros); check health and culture against public inventories before deciding on facilities; and where lower incomes already live close to everything, protect the density and public facilities that make it possible.

Closeness is not distributed by income but by density: where working-class neighborhoods are dense they have daily life nearby; where they are not, lower incomes are left far away.