#### Glasnik Matematicki, Vol. 51, No. 2 (2016), 447-452.

### CERTAIN WEAKLY GENERATED NONCOMPACT, PSEUDO-COMPACT
TOPOLOGIES ON TYCHONOFF CUBES

### Leonard R. Rubin

Department of Mathematics,
University of Oklahoma,
Norman, Oklahoma 73019,
USA

*e-mail:* `lrubin@ou.edu`

**Abstract.**
Given an uncountable cardinal *ℵ*,
the product space *I*^{ℵ}, *I=[0,1]*, is called a
Tychonoff cube. A collection of closed
subsets of a subspace *Y* of a Tychonoff cube *I*^{ℵ} that covers
*Y* determines a weak topology for *Y*.
The collection of compact subsets of *I*^{ℵ} that have a
countable dense subset covers *I*^{ℵ}.
It is known from work of the author and I. Ivanšić
that the weak topology generated by this
collection is pseudo-compact. We are going to show that it
is not compact. The author and I. Ivanšić have also
considered weak topologies on some other ``naturally occurring''
subspaces of such *I*^{ℵ}. The new information herein along with
the previous examples will lead to the existence of vast
naturally occurring classes of
pseudo-compacta any set of which has a pseudo-compact
product. Some of the classes consist of Tychonoff spaces,
so the product spaces from subsets of these are also Tychonoff
spaces.

**2010 Mathematics Subject Classification.**
54A10, 54B10, 54D30.

**Key words and phrases.** First uncountable ordinal space, products,
pseudo-compact, Tychonoff cube, weak topology.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.51.2.11

**References:**

- R. Engelking, General Topology, PWN-Polish
Scientific Publishers, Warsaw, 1977.

MathSciNet

- I. Ivanšić and L. Rubin,
*Pseudo-compactness
of direct limits*, Topology Appl. **160** (2013), 360-367.

MathSciNet
CrossRef

- I. Ivanšić and L. Rubin,
*The topology
of limits of direct systems induced by maps*, Mediterr. J.
Math. **11** (2014), 1261-1273.

MathSciNet
CrossRef

- I. Ivanšić and L. Rubin,
*Finite products of
limits of direct systems induced by maps*,
Appl. Gen. Topol. **16** (2015), 209-215.

MathSciNet
CrossRef

- I. Ivanšić and L. Rubin,
*Product theorems and examples in pseudo-compactness*, Asian J. of Math. & Computer Research **4** (2015), 16-23.

- R. M. Stephenson, Jr.,
*Pseudocompact spaces*,
Trans. Amer. Math. Soc. **134** (1968), 437-448.

MathSciNet
CrossRef

*Glasnik Matematicki* Home Page