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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.

57719 questions
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Some problems on locally finite sets

This is a series of problems on locally finite sets with a few I am stuck on. Can the community provide some help on how to proceed? Definition. Let $(X,\tau)$ be topological space. A set $\mathcal S$ of subsets of $X$ is said to be locally finite…
willyx888
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Are these two definitions of base for a topology equivalent?

I have some experience with metric spaces, but I never took a dedicated Topology class. I recently encountered the term "base for a topology", and looking on Wikipedia, there seem to be two definitions. The very first line of the article…
user56202
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Limit set of a function is a closed set

Let $f: M\rightarrow N$ be a continous function between topological spaces, in the case that I am interested we are working with manifolds so we have nice properties. We define the limit set of $L(f)$ as the set of $y\in N$ such that…
Someone
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Tiling $\mathbb{R}^2$ with polyomino

Divide $\mathbb{R}^2$ into disjoint unit squares, let Q be a tile (a polyomino, a finite set of connected unit squares). If, for every finite set S of unit squares in $\mathbb{R}^2$, - I can find a finite set of disjoint Q-tiles (a tiling) such that…
pi_yum_yum
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Does a subspace of R^2 being connected imply an epsilon neighborhood of that space is connected?

I'm working on a topology homework problem and a hint would be appreciated. The idea is, we have a connected subspace $A$ of $\mathbb{R}^{2}$ and we define $A_{\varepsilon}$ to be the set of all points in $\mathbb{R}^{2}$ that are within…
Peter A
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In a Hausdorff door space, at most one point is a limit point

I'm having some trouble proving the following: If $(X,\tau)$ is a Hausdorff door space, then at most one point $x \in X$ is a limit point of $X$. My approach was the following: I assumed that there existed $y \in X$ such that $y$ is also a limit…
Eduardo Magalhães
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Second countable scattered spaces are countable

Let $X$ be a second countable space and $X$ is scattered. A scattered space is a space for which every not empty subset has an isolated point. How to show $X$ is countable? Thanks ahead.
Paul
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A true/false questions on general topology

which of the following statements are true? ($a$) If every countable subset of a topological space is closed, then the space is discrete. ($b$) Every closed function from one space onto another is open. ($c$) Every discrete space is…
poton
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Continuous real-valued functions defined on non-compact spaces

A continuous real-valued function defined on a compact space is bounded and attains its bounds. Is it so that on a non-compact space it is always possible to define a continuous function that does not have these properties? (In fact, this is the…
Rodney Coleman
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Open map and topology

If $ f:\mathbb{R} \to \mathbb{R}$, $f(x)= x^3+3x^2+ax+3$, for what $a$ is $f$ an open map? I was thinking the following: It suffices to show $f$ maps basic open set (interval) to an open set in $\mathbb{R}$. Since $f$ is a continuous map because it…
user74074
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Is there a “universal” Hausdorff space?

Question. Is there a Hausdorff space $ X $ such that every Hausdorff space $ Y $ can be embedded into $ X^\Lambda $ (given the product topology) for some sets $ \Lambda $? Concerning Tychonoff spaces, $ X = [0, 1] $ satisfies the condition; i.e.,…
o-ccah
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Characterization of local compactness

k-spaces can be characterized as the topological inductive limit of compact spaces. Locally compact spaces are k-spaces but not conversely (k-spaces can also be characterized as quotients of locally compact spaces). Locally compact spaces have the…
yada
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Which spaces can be seen as to be the one-point compactification of some other space

Given a topological space $X$, under what conditions is $X$ the one-point compactification of some other space $Y$? Obviously $X$ must be compact but it seems unlikely to me that every compact space arises from one-point compactification.
1729
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Hausdorff spaces and continous functions on dense sets

I have a problem with an exercise regarding the equivalence of the Hausdorff property with a statement about continuous functions on dense sets. Let X be a topological space. Show that the following assertions are equivalent: a) X is a Hausdorff…
Polymorph
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Identification of the infinitely dimensional unit sphere

I have some doubts about a statement from class: Let $\mathbb{R}^\omega$ be the countable product of $\mathbb{R}$. Equip $\mathbb{R}^\omega$ with the scalar product $\sum_{k = 1}^\infty x_i y_i$ and the induced norm. We identify $\mathbb{R}^n$ with…
mathology
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