Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,448
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Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
- 395
2
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1
answer
187
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Preservation of topological properties in between two topologies
Let $X$ be a set, $\tau_1 \leq \tau_2$ two comparable topologies on $X$ ($\tau_1$ is weaker than $\tau_2$) and consider some topological property $\varphi$ that holds for both $\tau_1$ and $\tau_2$.
I ...
- 1,741
2
votes
1
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154
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Reconstructing relations with the image relation of a topology
For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R_{im}(X,\tau)$ is reflexive. This ...
- 45.5k
2
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2
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412
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Is an open subset of a compact subset of a Hausdorff locally convex TVS paracompact?
This repeats the title in a more readable way. Take a compact subset $X$ of a Hausdorff locally convex topological vector space and $U$ be an open subset of $X$. Is $U$ paracompact?
2
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1
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238
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$T_2$-space $X$ with $X\cong \text{Aut}(X)$
Is there an infinite $T_2$-space $X$ with $X\cong \text{Aut}(X)$? (Here, $\text{Aut}(X)$ is the set of automorphisms $\varphi:X\to X$ and it carries the topology inherited from the product topology on ...
- 45.5k
2
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1
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232
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Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space?
Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space? (You find the definition of $\mathcal{P}(\omega)/fin$ here.)
Remark: According to this, the interval topology of $\mathcal{P}...
- 45.5k
2
votes
2
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339
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Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?
Definitions and notations.
Let $\mathcal{P}(X)$ the power set of $X$.
Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X.
We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...
- 163
2
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1
answer
135
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A souped-up version of a question asked previously about uncountable subsets of topological spaces
Let T be an uncountable Hausdorff space. The following property of T will be referred to as "property P". If S is any uncountable subset of T, then the set of all points of S that are not limit points ...
- 6,105
2
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1
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258
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Topological degree of homogeneous function of degree k [closed]
Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k?
This is true if F is ...
- 81
2
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1
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119
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Actions of compact Lie groups on (possibly but hopefully not) non-regular spaces
Suppose $G$ is a compact Lie group acting freely on a topological space $Q$ (about whose separation conditions I make no assumptions) and the qoutient $Q/G$ is known to be completely regular Hausdorff ...
- 15.3k
2
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1
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627
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A uniformity with a countable base is a pseudometric uniformity.
I need a proof for this proposition:
If a uniformity $\mathfrak U$ on $X$ has a
countable fundamental system of
entourages, then it can be defined by
a pseudometric on $X$.
which is the ...
- 31
2
votes
1
answer
1k
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Free and cellular G-action implies free G-complex?
Recall that a CW-complex $X$ with an action of a group $G$ which permutes the cells (i.e., for any $g \in G$ and any cell $\sigma \subseteq X$, $g\sigma$ is a cell) is called a $G$-complex. If the ...
- 21
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1
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1k
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Is every connected regular space having more than one point uncountable?
Is every connected regular space having more than one point uncountable?
- 21
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671
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Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
- 2,318
2
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2
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764
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On a special case of Alexander duality
Let $S^n$ be the $n$-dimensional sphere and let $K\subseteq S^n$ be a compact, locally contractible subspace of real codimension $\geq 2$. Applying Alexander duality we find that
$$
\tilde{H}_{i}(S^n-...
- 7,521
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2
answers
645
views
Varieties, Frechet Completions, and Regular Functions
Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
- 1,442
2
votes
1
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583
views
Is the Hopf link a Brunnian link?
I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on links). Not wishing to write ...
- 26.4k
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3
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1k
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Countable atomless boolean algebra covered by a larger boolean algebra
Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There ...
- 345
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247
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Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?
As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
- 1,141
2
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1
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217
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Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?
Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology:
The initial topology with respect to the family maps $(\...
2
votes
1
answer
139
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Variation of concept of a Lusin space
Citing from Wikipedia,
A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space.
Is there a (previously studied) analogous concept of a Hausdorff (...
- 611
2
votes
1
answer
276
views
Global control of locally approximating polynomial in Stone-Weierstrass?
Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\...
- 451
2
votes
1
answer
218
views
Existence of diffeomorphism interpolating affine map and identity
$\newcommand{\R}{\mathbb{R}}$Suppose $\Omega$ is a bounded, convex domain in $\R^{m}$. Fix $x_1, x_2\in\Omega$ and an invertible matrix $A\in\mathrm{GL}^{+}(m)$ with positive determinant.
Let $U\...
- 193
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1
answer
178
views
A stronger version of paracompactness
Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...
- 755
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votes
1
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340
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$4$-manifold with simply connected boundary
This may be a very silly question but I could not get any counter-example.
Let $M$ be a compact differential $4$-manifold with boundary $dM$.
Suppose that the inclusion map induced map $\pi_1(dM) \to \...
- 1,059
2
votes
1
answer
287
views
(Homotopy) colimit and manifold
Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...
- 37
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votes
1
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254
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Special version of $\Delta$-system Lemma for singular cardinals
In his article "Remarks on cardinal invariants in topology" (you can get the paper here: Where can I find the following S. Shelah's paper?), Saharon Shelah states the following claim:
(...
- 622
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1
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155
views
Is $\mathbb R$ with cocountable topology star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 475
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2
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469
views
What to call a continuous function with preimage preserving nowhere-density?
Currently I am reading some basic literature on descriptive set theory and boolean algebras. And this comes out a lot, for example in results like:
Let $X$ and $Y$ be topological spaces, and $f:X \to ...
2
votes
1
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249
views
An example of a $T_1$ space where all closed $G_\delta$ sets are zero-sets, but it isn't normal
In Engelking's General topology, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:
A $T_1$ space is normal iff the following properties hold (both):
Every closed $...
2
votes
1
answer
129
views
Approximate Jordan-Brouwer theorem
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is ...
- 5,385
2
votes
1
answer
200
views
A variation of closed-subgroup theorem
$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...
- 14k
2
votes
2
answers
313
views
Construct a homeomorphism whose periodic points set is not closed
I'm looking for a simple example in discrete dynamical systems whose periodic points set is not necessary closed.
I've seen some example in websites but they are not that simple and discrete.
Note ...
- 145
2
votes
2
answers
126
views
Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$
Consider Cantor space $2^\omega$ with the standard topology generated by open sets $[\sigma] = \{ \sigma^\frown x: x \in 2^\omega \}$. If $A \subseteq 2^{<\omega}$ and $x \in 2^\omega$, we say $A$ ...
2
votes
2
answers
395
views
Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
- 4,969
2
votes
1
answer
178
views
Are open sets determined by paths?
Let $X$ be a topological space, let $U \subset X$, and suppose that for every path $\gamma\colon [0,1] \to X$ the preimage $\gamma^{-1}(U)$ is open. Is it true that $U$ is open? Presumably not in ...
- 641
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2
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309
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Properties of the topology of sequential convergence $\tau_\text{seq}$
Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_\text{seq}$ has the ...
- 1,245
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110
views
Is there a locally countable and weakly Lindelöf space which is not ccc
Is there a locally countable and weakly Lindelöf space which is not ccc?
A space $X$ is locally countable if for each point $x\in X$ there is an open neighbourhood $O_x$ of $x$ such that $|O_x| < \...
- 63
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73
views
Are compactifications of completely $T_{4}$ spaces completely $T_{4}$?
The title is the question.
Given a locally compact completely $T_{4}$ space $X$ (every subspace is $T_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely ...
- 213
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2
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308
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The space of Borel function modulo comeager sets is Dedekind complete
Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...
- 267
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votes
1
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89
views
$T_1$-spaces vs $T_1$-hypergraphs
Let us say that a hypergraph $H=(V,E)$ is $T_1$ if for $x\neq y$ there is $e\in E$ such that $e\cap\{x,y\} = \{x\}$.
Note that for any $T_1$-space $(X,\tau)$ the topology $\tau$ contains the ...
- 45.5k
2
votes
1
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115
views
Non-discrete $T_2$-space $(X,\tau)$ with $2^{|X|}$ retracts
If $(X,\tau)$ is a topological space, we call $A\subseteq X$ a retract if there is a continous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$ (we assume $A$ to be endowed with the subspace ...
- 45.5k
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1
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184
views
Under what conditions the End-compactification is metrizable
Suppose that $X$ is a hemicompact space, connected and locally connected. In that case, it seems that it is possible to define a "End-compactification" of $X$ (in the sense of Freudenthal).
Suppose ...
- 975
2
votes
1
answer
208
views
Relation between the weak star topology and hereditary Lindelöfness
Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$
The converse is clearly true, since the ...
- 3,992
2
votes
2
answers
484
views
covering theory with compact open topology
In the following all spaces $C^0(X,Y)$ are spaces of base point preserving maps with the compact-open topology.Furthermore all spaces I consider in the following are locally pathwise connected.
Under ...
- 475
2
votes
1
answer
286
views
Embedding into $C\times [0,1]$
Every totally disconnected separable metric space of dimension $n$ homeomorphically embeds into $C\times \mathbb R ^n$.
Is something like this known? $X$ is totally disconnected means that every ...
- 3,055
2
votes
1
answer
195
views
Non-uniqueness in Krylov-Bogoliubov theorem
So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a ...
- 24.6k
2
votes
2
answers
694
views
"Minkowski Multiplication" of Convex Sets?
Apologies if this question might be trivial or has been asked already (haven't found an equivalent post), but I am trying to figure out whether the following is true:
Given two convex sets $\mathcal{...
- 123
2
votes
1
answer
349
views
Why is an inductive limit of bornological spaces bornological?
Let $(E_\alpha,\tau_\alpha,g_\alpha)$ be a family of bornological (locally convex) topological vector spaces $(E_\alpha,\tau_\alpha)$, where a LCTVS $E$ is said to be bornological if every circled, ...
- 1,207
2
votes
1
answer
555
views
Could Furstenberg's Argument Prove the Infinitude of Primes in Number Fields?
I have a somewhat unconventional view of the Prime Number Theorem as a "quantification" of the infinitude of primes. Here I recall the argument of Furstenberg. Define a topology $\mathcal{X}$ on $\...
- 22.6k