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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
1
vote
Example of a completely regular spaces
Partial answer: if the space is zero-dimensional then any two disjoint open sets have disjoint closures, so the space is extremally disconnected. And then closures of open sets are clopen, hence your …
8
votes
When factors may be cancelled in homeomorphic products?
How about Bing's example: The cartesian product of a certain nonmanifold and a line is E4. The `other' factor is the dog-bone decomposition of three-space.
3
votes
Accepted
Connectedness of a union regading a proximity
Consider X∩A and Y∩A, starting from a partition {X,Y} of A∪B. If both intersections are nonempty we are done, as (X∩A)δ(Y∩A). Otherwise, A⊆X, …
8
votes
Accepted
Is there a notion of a "perfectly regular" topological space?
An answer is: completely regular plus countable pseudocharacter, the latter means that points are Gδ-sets. In completely regular spaces a point is a Gδ-set iff it is the zero-set of a …
4
votes
A question about disconnecting a Euclidean space or a Hilbert space
Assume the complement of S in Rn is not connected, say A and B are relatively closed and disjoint in Rn∖S (and nonempty of course); let O be the complement of …
5
votes
Characterization of Tychonoff spaces in terms of open sets
Another answer was given by Aarts and De Groot in Complete regularity as a separation axiom Canadian Journal of Mathematics 21 1969 96–105. Frink's condition sets things up for a proof a la Urysohn's …
2
votes
Accepted
zeroset-diagonal
All it takes is a countable, not first-countable, Tychonoff space, say a countable dense subset, D, of the Cantor cube 2c. For every point (d,e) off the diagonal there is a continu …
4
votes
Accepted
Could IX be seen as a subspace of IβX under the compact-open topology?
The two sets are essentially the same: the map that sends every f∈IβX to its restriction is a bijection; the two topologies are, in general, not the same. The compact-open topology on $I^ …
3
votes
Accepted
Infinite closed partition of the real line with no closed infinite unions
Let F be this partition. As noted above it must be uncountable. We may as well assume it lives on the interval (0,1) and add the set {0,1} to each member to obtain a fam …
3
votes
Accepted
Metrizable implies hemicompact
For regular spaces the implication is false in general: let A be regular and such that all continuous real-valued functions on it are constant (see Problem 2.7.17 in Engelking's General Topology). T …
5
votes
Accepted
closed subset of weakly lindelof
The Niemytzki plane is weakly Lindelöf (the open upper half plane is a dense Lindelöf subspace); the x-axis is an uncountable closed and discrete subspace.
4
votes
Accepted
Counterexample about Jones lemma with special weak condition.
Yes, take, for example the Sorgenfrey plane P. A standard example of a non-normal space. It is separable and its anti-diagonal {(x,−x):x∈R} is closed and discrete, so Jones …
0
votes
Accepted
F-spaces and points whose complements are C*-embedded
Consider the F-space ω∗ (the Cech-Stone remainder of ω).
Under the Continuum Hypothesis there is no point whose complement is C∗-embedded. On the other hand it is also consistent …
2
votes
Bases of completely regular (Tychonoff) spaces
Here's a counterexample to 1.
Let T be the Tychonoff plank, i.e., the product
(ω1+1)×(ω+1) with the point ⟨ω1,ω⟩
removed.
Consider the set $\omega\times …
3
votes
Large discrete subsets of connected T2-spaces
If you want a closed discrete set take the metric hedgehog with κ many spines, where κ is the desired cardinality. If κ<c replace [0,1] by a countable connected Hausd …