# Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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67 views

### Kastanas' game and completely Ramsey sets

recently I was reading the article ''On the Ramsey property for sets of reals'' of Ilias Kastanas (https://www.jstor.org/stable/2273667?seq=1#metadata_info_tab_contents), in this article the author ...

**3**

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**1**answer

116 views

### A pair of spaces equivalent to a pair of CW-complexes

Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that
$Z-A$ is homeomorhic to $X-Y$ and
$Z/A$ homeomorphic to $X/Y$ and
The closure of $Z-A$ ...

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218 views

### “a result of Atiyah implies that the top cell of an orientable manifold splits off stably”

I am looking for a reference for the following sentence
“a result of Atiyah implies that the top cell of an orientable
manifold splits off stably”
Thank you very much for helping me find a reference ...

**1**

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47 views

### What are the various kinds of graphs that can be defined on $C(X)$

I was considering the space $C(X)$ where $X$ is a topological space and $C(X)$ is the set of all continuous functions from $X$ to $\Bbb R$.
What are the various kinds of graphs that can be defined on ...

**5**

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105 views

### Closed embedding of CW-complexes

Suppose that $i: X\rightarrow Y$ is a closed embedding such that $X$ and $Y $ are (retracts) of CW-complexes. Does it follow that $i$ is a cofibration ?
Remark: There is a similar question here, ...

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**1**answer

58 views

### Filtered colimit of a topological space

Suppose that $X$ is a space filtered by closed subspaces $X_{1}\subset X_{2}\subset \dots$.
As topological space $X=\operatorname{colim}_{n}X_{n}$.
We define $Y_{n}=X_{n+1}/X_{n}$, and consider the ...

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21 views

### Constructible subspaces of sober spaces

Let $X$ be a topological space such that every closed irreducible subset has a unique generic point. We know that locally closed subspaces of $X$ also have this property. Is this true for ...

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241 views

### Restriction of a cofibration to closed subspaces

Let $i: X\rightarrow Y$ be a cofibration between CW-complexes, more precisely a cellular embedding. Let $A$ be a closed subspace of $Y$ and $Z=i^{-1}(A)$. Let $$j: Z\rightarrow A$$ be the restriction ...

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**1**answer

91 views

### Continuity of the Restriction Map Between Function Spaces [on hold]

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,...

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100 views

### When does a function space with compact-open topology have countable chain condition?

As in title，when a function space with compact-open topology has countable chain condition？ Are there some sufficient and necessary conditions？ Who give some references about this topic？
McCoy and ...

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53 views

### Dimension of a topological space equals the supreme of the dimension of its open cover

For a topological space $X$ which is covered by a family of open subsets $\{U_i\}$, then show that $\dim(X)=\sup (\dim(U_i))$.
I understand that $\dim(X)\geq \sup(\dim(U_i))$, so it only suffices to ...

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83 views

### Quotient space homeomorphic if the complements are homeomorphic? [closed]

Let $A_{0}$ be a closed subspace of $A$ and $B_{0}$ a closed subspace of $B$ such that there exists a
$i:A\rightarrow B$ a continuous injective map (or even a continuous embedding) such that the ...

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109 views

### What does the Grothendieck topos tell us about the homotopy type of a space?

Let $M_1$, $M_2$ be two closed connected topological manifolds. We can consider the small sites of open coverings of them, and the categories of sheaves on these sites.
what can we say about $M_1$ ...

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29 views

### Problem on Topological Spaces [closed]

I have to solve this problem, in particular the point of the union of an arbitrary family of elements that belongs to the topology.
This is the Text: "Let {p} be an arbitrary singleton set such that ...

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70 views

### Connectedness of sequence spaces (countable products) in different metrics

My question concerns a quite elementary problem in set-theoretic topology: Assume that $(X,d)$ is a compact metric space. Consider the infinite product $X^{\mathbb{Z}}$ of all (two-sided) sequences in ...