# Questions tagged [gn.general-topology]

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

2,951 questions
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### Relatively countably compact subsets without countably compact closure.

I am looking for a (Hausdorff or better) space $X$ and a subset $A$ of $X$ that is relatively countably compact (every sequence from $A$ has an accumulation point in $X$) such that the closure of $A$ ...
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### Non-homeomorphic spaces that have continuous bijections between them

What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
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### Union of proximally connected sets

Let (δ;U) is a proximity space. I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y. Is the following true? (I need a proof or a counter-example.) Conjecture If S ...
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### Definition of Connected Subspace

In Munkres (Chapter 3, Section 23, p. 148), Munkres shows that if a subspace $Y$ of a space $X$ is not connected, then there are two disjoint open subsets $A,B$ such that the union of $A$ and $B$ ...
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### The Wedge Sum of path connected topological spaces

A definition of wedge sum can be found here: http://en.wikipedia.org/wiki/Wedge_sum My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy ...
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### What is the pure intuition for topological continuity and topology? [closed]

I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity. The ...
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### profinite spaces coming from profinite groups

This is probably well-known: Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed? - Is every profinite group ...
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### motivation for compactness [duplicate]

Possible Duplicate: How to understand the concept of compact space Hello, I am learning some analysis on my own and what is the motivation to consider compactness? eg. I do not understand why ...
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### Countable open subgroup

In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
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### Functoriality of base change

Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$...
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### Is there a notion of a “perfectly regular” topological space?

The separation axioms have exploded a little since the original list of four! Amongst them can be found "completely regular" spaces and "perfectly normal" spaces. The former is well-known: a point ...
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### More on continuous images of dense orders

In this question I asked if there was an analogue of connectedness which applied to dense orders which were not required to be complete. Between them, Noah and Joel showed that every (infinite) ...
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### When can boundedness be characterized topologically in Metric spaces?

Let H be a separable and infinite-dimensional Hilbert space. Is every closed subset of H homeomorphic to some closed and bounded subset of H?
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### What, if anything, can be said about continuous images of densely ordered spaces?

If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove ...
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### Applications of compactness [closed]

Similar to this question: Applications of connectedness I want to collect applications of compactness. E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
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### profinite spaces are the pro-completion of finite sets

The title sounds tautological, right? Perhaps I'm missing something completely trivial here ... Assume $X$ is a compact totally disconnected hausdorff space. It is known that $X$ can be written as ...
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### Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]

My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitary metric space (which makes it plausible to me that metric spaces are ...
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### Connectedness and the real line

It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be ...
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### Besicovitch Covering Constant for R^1

In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover. The Besicovitch Covering ...