[METRIC SPACE]
[CAUCHY SEQUENCE, COMPLETENESS
AND CONTRACTION PRINCIPLE]
[REAL NUMBERS AS A COMPLETE
ORDERED FIELD]
[DENSE SUBSETS, BAIRE-CATEGORY
THEOREM]
[SEPARATE, COUNTABLES SPACES AND
CONTINUOUS FUNCTIONS]
[COMPACTNESS]
[CONTINUOUS FUNCTIONS, COMPACT SETS AND CONNECTEDNESS]