Dan Ma's Topology Blog

We discuss a topological characterization of Cantor sets. Topologically speaking there is only one Cantor set. The topological characterization consists of these three properties: ….….zero-dimensional, ….….perfect, ….….compact metric space. Thus, every space satisfying this characterization is homeomorphic to the Cantor … Continue reading →

More eye opening information from Gemini. A post called A confession by Google AI Gemini (see here) was published. It has now come full circle: I actually got to engage Gemini in a Google chat window about that post and … Continue reading →

We discuss whether several separation axioms are preserved under closed maps. Consider the following diagram. Diagram 1 – Separation Axioms …. ….……….Perfectly normal …………………… ….……….Hereditarily normal …………………… ….……….Normal …………………… ….……..Completely regular …………………… ….……….Regular …………………… ….……….Hausdorff …………………… ….……….Every one-point set is … Continue reading →

This is a follow up to a recent piece about my experience with Gemini repsect to a topology query (see here). In the previous query, Gemini provided results that were wildly inaccurate. In the follow up query, I managed to … Continue reading →

This is a discussion of the search results from Google AI, which appear to be questionable. To see what information I can find on the Internet on a particular class of spaces, I entered the following query in Google. See … Continue reading →

The set of all ultrafilter on equipped with the Stone topology is a compact space containing as a dense subset. In fact, this space of ultrfilters on is , the Stone-Cech compactification of , up to equivalence (see here). In … Continue reading →

It is well known that there are countably compact spaces whose product is not countably compact. We present examples that are subspaces of the Stone-Cech compactification of the countable discrete space . We construct countably compact such that and is … Continue reading →

It is well known that the intersection of countably many open subsets of the Stone-Cech remainder has non-empty interior (see here). It follows from this fact that the Stone-Cech remainder has no point-countable -bases. We give a proof of this … Continue reading →

We discuss one more basic fact of , the Stone-Cech compactification of the countable discrete space . We show that any non-empty -subset of the remainder has non-empty interior. We then make a brief comment on P-points of . In this discussion, we continue to treat as the space of ultrafilters on with the Stone topology (see here for an introduction). This article is part of a series of articles o…

We show that the Stone-Cech compactification of the countable discrete space can be described as the space of ultrafilters on with a topology called the Stone topology. The space of all ultrafilers on is a compact Hausdorff space with a countable dense subset that is a homeomorphic copy of and is thus a compactification of . In addition, this compactification satisfies a function extension proper…

Given a completely regular space , is the Stone-Cech compactification of the space . We present three spaces whose Stone-Cech compactifications can be described explicitly. That is, these examples of are familiar spaces. For these examples, we can make such determination because a completely regular space is C*-embedded in its Stone-Cech compactification and any compactification of in which is C*…

We prove that , the Stone-Cech compactification of the integers, is not hereditarily normal by providing an example of a non-normal subspace. To define this example, we make use of the fact that the remainder contains continuum many pairwise disjoint open subsets. The example is found in the Engelking text [Examples 3.6.18 and 3.6.19, 1] This article is part of a series of articles on Stone-Cech …

Consider , the Stone-Cech compactification of the countable discrete space . It has no non-trivial convergent sequence. As a result, it fails to have any convergence properties that require the existence of non-trivial convergent sequences. For example, is not a sequential space (see Fact 7 found here). Any sequential space has countable tightness (see here). We show that even fails to have count…

We list all the separable but not Lindelof spaces found in Counterexamples in Topology by Steen and Seebach [2]. There are ten such examples. We define the examples and list out the key facts. On these examples, we particularly single out separability and Lindelofness. See here for examples of spaces that are Lindelof but not separable. The disxcussion here is an attempt to locate examples with t…

We discuss Novak Space. This is Example 112 in Counterexamples in Topology by Steen and Seebach [3]. Novak Space is a classic example of a countably compact space whose square is not countably compact. Novak space is a subspace of , the Stone-Cech compactification of a countable discrete space. Our goal is to work out all the details, as described in [3], and fill in the gaps where necessary. Exa…

It is quite easy to derive a space that failes to have a countable base at one point. We can take a metric space and collapse a set to one point. The resulting quotient space may fail to be first countable at the identified point but is still first countable at every other point. See here and here for such an example. This is a famous quotient space called the sequential fan. Another famous examp…

It is well know that countably compactness plus a covering property means that the space is compact. For example, it follows from definitions that any Lindelof countably compact space is compact. Because every locally finite collection of sets in a countably compact space is finite (see Theorem 3 found here), paracompact + countably compact implies compact. Furthermore, any metacompact countably …

We present a non-metrizable compact space called Alexandroff Square, which is Example 101 in Steen and Seebach [4]. This is a compact space that is due to Alexandroff and Urysohn, which can be traced back to their paper published in 1929. One of the facts duscussed in [4] is that the Alexandroff Square is not first countable at any point on the diagonal. We show that the Alexandroff Square, thoug…

A space is normal if for every disjoint closed subsets and of , there exist disjoint open subsets and of such that and . In other words, in a normal space, every two disjoint closed subsets can be separated by … Continue reading →

A space is Lindelof if every open cover has a countable subcover. A space is separable if it has a countable dense subset. We present examples of Lindelof spaces that are not separable. The Lindelof property and separability are countability … Continue reading →