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# Questions tagged [gn.general-topology]

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

3,057 questions
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 ...
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 $Z-A$ ...
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 ...
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 ...
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, ...
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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### 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 ...
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 ...
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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### 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 ...
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 ...
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 ...
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 $M_1$ ...
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 ...
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 ...

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