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Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

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Continuity of $f:X\to [0,1]$ where $X$ is homeomorphic to the Cantor set.

This is an exercise from Mendelson's Introduction to Topology, page 101. THEOREM For each $n\in \Bbb N$, let $X_n$ be the discrete two-point topological space $\{0,2\}$. Define the product space $X=\prod\limits_{n\in \mathbb N}X_n\;$. Let $f:X\to…
Pedro
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Finite Number of Partitions of Unity in a Compact Hausdorff Space

I'm working on this proof in Gamelin "Introduction to Topology" and I think I'm almost at the result, I'm just a little stuck with how to proceed. It is this. Let $X$ be a be compact Hausdorff space and let {$U_\alpha$}$_{\alpha \in A}$ be an open…
Leif Ericson
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Show that any non-empty, non-compact convex subset of $\mathbb{R}^n$ lacks the fixed point property

The Fixed-Point Property is stated as such: Every continuous self-map admits a fixed point. Attempt: Since compact sets in $\mathbb{R}^n$ are necessarily closed and bounded, we consider two cases. Where the subset is non-closed and when the set is…
Mathemphetamine
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The boundary of set is the set itself examples

Could you provide me some examples of sets, which are not based on Cantor's construction, that satisfy the property $\partial A=A$, that is the boundary of a set is the set itself?
Salech Alhasov
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A non-Hausdorff space with unique limit

Can I find a topological space $X$ such that every convergent sequence in $X$ has a unique limit in $X$, but $X$ is not Hausdorff?
Silent
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If $f: \mathbb{R}\setminus\mathbb{Q} \to \mathbb{R}_S\times\mathbb{R}_S$ is continuous, then the image has empty interior.

Please, don't write the entire answer. I am looking for hints only. Here $\mathbb{R}_S\times\mathbb{R}_S$ is the Sorgenfrey plane. My attempt so far was limited by Suppose it is continuous and it has a non-empty interior. Let $p$ be an interior…
B. Rivas
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Further examples of 1-transitive but not 2-transitive groups of self-homeomorphisms

In a question from last week, I was searching for "nice" spaces on which the group of self-homeomorphisms acted $1$-transitively but not $2$-transitively on the space. In that case, I had the condition that the space be connected and Hausdorff. The…
Thomas Andrews
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Connected nice topological spaces that are transitive but not 2-transitive under homeomorphisms

I was thinking about a question on here earlier, and came up with this question. [Added Hausdorff note, below.] It is easy to see that the group of self-homeomorphisms of the real line acts $2$-transitively on the space, but not $3$-transitively.…
Thomas Andrews
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Does every topology have a subbasis?

I know that every topology is generated by a basis. Is it true that every topology has a subbasis which generates it? If not, what makes it not possible to "synthesize" a subbasis out of a basis? Having studied linear algebra, I am intuitively…
Jennifer
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Topology of the power set

Does anyone know a non trivial(that we cannot define on every set) topology defined on the power set of an uncountable set?
t.k
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Intuition under Neighbourhood concept in Topological Spaces

I sense that I didn't grasp well the concept of “Neighbourhood” in topological spaces. I did read through definitions of neighborhood, open and closed spaces. Despite that, I miss the utility of such a concept. For example, in the definition of…
RanCat
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Closure of Interior and Interior of Closure

I know questions similar to this have been asked here but, is it possible to find a subset of a topological space such that its closure of interior and interior of closure does not contain each other? For example if $X=\mathbb{R}$, $A=\mathbb{Q}$,…
verticese
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Expansion lemma for paracompact hausdorff space.

[Munkres, Section41, Ex5] Let $X$ be paracompact. We proved a "shrinking lemma" for arbitrary indexed open coverings of $X$. Here is an "expansion lemma" for arbitrary locally finite indexed families in $X$. Lemma. Let $\{ B_ \alpha \} _ {…
Gobi
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Every regular , T_1 space is a Urysohn space

Definition: A space $X$ is a Urysohn space iff whenever $x \neq y$ in $X$ there are nhoods of $U$ of $x$ and $V$ of $y$ such that $\overline{U} \cap \overline{V} = \emptyset$. I want to show that every regular, T_1 space is Urysohn. My attempt: Let…
user
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Is $\mathbb{R}^\omega$ a completely normal space, in the box topology?

Basically, what the title says. Is $\mathbb{R}^\omega$ a completely normal space in the box topology ? ($\mathbb{R}^\omega$ is the space of sequences to $\mathbb{R}$) Thanks !
thetruth
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