Retractions and other continuous maps from onto

Author:
W. W. Comfort

Journal:
Trans. Amer. Math. Soc. **114** (1965), 1-9

MSC:
Primary 54.53

DOI:
https://doi.org/10.1090/S0002-9947-1965-0185571-9

MathSciNet review:
0185571

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Our two main theorems are stated below. The first is proved with the aid of the continuum hypothesis.

Theorem 2.6. [CH] *Suppose that there is a retraction from* *onto* . *Then X is locally compact and pseudocompact*.

Theorem 4.2. *Let D be a discrete space whose cardinal number m exceeds* 1. *In order that there exist a continuous function from* *onto* , *it is necessary and sufficient that* .

The proof of Theorem 2.6 rests on a result of Walter Rudin concerning *P*-points (see 1(d) and 1(e) below); Theorem 4.2 depends on the following simple result, which appears to be new.

Theorem 4.1. *Let D be the discrete space with cardinal number* . *The smallest cardinal number which is the cardinal number of some dense subset of* *is* .

**[1]**W. W. Comfort and H. Gordon,*Disjoint open subsets of*, Trans. Amer. Math. Soc.**115**(1964), 513-520. MR**0163280 (29:583)****[2]**L. Gillman and M. Henriksen,*Concerning rings of continuous functions*, Trans. Amer. Math. Soc.**77**(1954), 340-362. MR**0063646 (16:156g)****[3]**L. Gillman and M. Jerison,*Rings of continuous functions*, Van Nostrand, Princeton, N.J., 1960. MR**0116199 (22:6994)****[4]**I. Glicksberg,*Stone-Čech compactifications of products*, Trans. Amer. Math. Soc.**90**(1959), 369-382. MR**0105667 (21:4405)****[5]**E. Hewitt,*Rings of real-valued continuous functions*. I, Trans. Amer. Math. Soc.**64**(1948), 45-99. MR**0026239 (10:126e)****[6]**M. Katětov,*On real-valued functions in topological spaces*, Fund. Math.**38**(1951), 85-91. MR**0050264 (14:304a)****[7]**J. L. Kelley,*General topology*, Van Nostrand, New York, 1955. MR**0070144 (16:1136c)****[8]**E. Marczewski,*Séparabilité et multiplication Cartésienne des espaces topologiques*, Fund. Math.**34**(1947), 127-143. MR**0021680 (9:98b)****[9]**W. Rudin,*Homogeneity problems in the theory of Čech compactifications*, Duke Math. J.**23**(1956), 409-419, 633. MR**0080902 (18:324d)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54.53

Retrieve articles in all journals with MSC: 54.53

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1965-0185571-9

Article copyright:
© Copyright 1965
American Mathematical Society