Skip to main content
Search type Search syntax
Tags [tag]
Exact "words here"
Author user:1234
user:me (yours)
Score score:3 (3+)
score:0 (none)
Answers answers:3 (3+)
answers:0 (none)
isaccepted:yes
hasaccepted:no
inquestion:1234
Views views:250
Code code:"if (foo != bar)"
Sections title:apples
body:"apples oranges"
URL url:"*.example.com"
Saves in:saves
Status closed:yes
duplicate:no
migrated:no
wiki:no
Types is:question
is:answer
Exclude -[tag]
-apples
For more details on advanced search visit our help page
Results tagged with
Search options not deleted user 5903

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 …
KP Hart's user avatar
  • 11.4k
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.
KP Hart's user avatar
  • 11.4k
3 votes
Accepted

Connectedness of a union regading a proximity

Consider XA and YA, starting from a partition {X,Y} of AB. If both intersections are nonempty we are done, as (XA)δ(YA). Otherwise, AX, …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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 RnS (and nonempty of course); let O be the complement of …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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 fIβX to its restriction is a bijection; the two topologies are, in general, not the same. The compact-open topology on $I^ …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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.
KP Hart's user avatar
  • 11.4k
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):xR} is closed and discrete, so Jones …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k
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 …
KP Hart's user avatar
  • 11.4k

1
2 3 4 5
12
15 30 50 per page