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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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Example of a paracompact space that is not metrizable

I'm looking for an example of a space which is paracompact but not metrizable. The definition of paracompactness that I'm working with is that $(X,\tau)$ is paracompact if it is Hausdorff ($T_{2}$) and for every open cover there exists a locally…
T. Eskin
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Invariance of domain in $\mathbb{R}^2$

Let $U \subseteq \mathbb{R}^2$ an open subset and let $f:U\rightarrow \mathbb {R}^2$ is be a continuous function. I have the following version of Invariance of Domain Theorem (in $\mathbb{R}^2$): If $f$ is injective then $f$ is homeomorphism. I…
alex kur
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Break up $\mathbb{R}P^2$ into a part homeomorphic to Mobius band & part homeo. to the 2-disc

The claim is that $\mathbb{R}P^2 = A \cup B$ where $A \simeq$ Mobius band, $B \simeq D^2$, and $A \cap B \simeq S^1$. I understand this intuitively with a gluing type argument, similar to the arguments here…
Schwinger
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Prove that if $(X,d)$ is a compact metric space, and $K$ is an infinite set in $(X,d)$, then if $K$ has no limit point, $K$ is a closed set.

Prove that if $(X,d)$ is a compact metric space, and $K$ is an infinite set in $(X,d)$, then if $K$ has no limit point, $K$ is a closed set. Idea : Just like most topology proofs, the way I want to approach this problem is to show that $X - K$ is…
Nick Freeman
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Prove that y>2x+1 is open?

The answer is inserted but what I'm looking for is a heavy breakdown on this. My professor tried to explain to me a harder version but I don't understand it. Solution: What my prof does for the problems he has gone over is he first draws out a…
August
  • 1,562
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A first-countable, compact space which is non-separable

Let $X$ be a space which is first countable and compact. Is $X$ necessarily separable? Is $X$ necessarily second countable?
Keith Penrod
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Topological Group Structure on a Subset of $\mathbb R$

I have been pondering over this question for quite a long time. Please help. Is it possible to define some group operation on $[a,b]$ (with the usual topology inherited from $\mathbb R$) so that it becomes a topological group? Thanks for any help.
Ester
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Show that $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=x+y$ is continuous using open sets.

The problem statement is, Show that $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=x+y$ is continuous using open sets. I know that to show $f$ to be continuous I take an arbitrary open set $O\subset\mathbb{R}$ and show that $f^{-1}(O)$ is open in…
S.D.
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How to cover an open subset of $\mathbb{R}^n$ with balls?

I am trying to solve the following exercise: If $U$ is an open subset of $\mathbb{R}^n$, show that there exists an increasing sequence $\{A_k\}^\infty_1$ of compact contented sets such that $U=\bigcup^\infty_{k=1}\ \mathrm{int}\ A_k$. Hint: Each…
koletenbert
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A characterisation of nowhere dense sets

Can you give me a hint, why the following characterisation of a nowhere dense set $A$ (where my definition of n.d. is $ \textrm{Int(cl}(A))=\emptyset$) should hold: For all nonempty open set $U$ there exists an open set $V$, such that $V \subseteq…
temo
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A question about homeomorphisms

If f is a real-valued function defined on a connected open set of real numbers, and f is injective and continuous, is the inverse of f continuous? I realize that in general the inverse is not necessarily continuous, but I believe that in this case…
Hexagon Tiling
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Show that any convex subset of $R^k$ is connected

I need to prove that any convex subset of $R^k$ is connected. I have seen the proof in Rudin's book and on numerous websites but they all use some prior results. I want to do it without using results where one establishes something for some other…
Silver moon
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Set of limit points of topologist's sine curve $S$

Let $S=\{(x,\sin(1/x)):x \in (0,1]\}$ be the topologist's sine curve. Find the limit points $\lim S$ of $S$. I claim $\lim S = S \cup \{(0,y):y \in [-1,1]\}$. But, how do you show that any of these points is such? Certainly there is some nice lemma…
JustAskin
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Open and Closed Sets definition

Does the complement of a set being closed necessarily imply that the set itself is open? Could the set be both/neither open and/nor closed if its complement is strictly closed?
Andrew
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Continuous map $\mathbb{S}^n\to \mathbb{S}^m$

Is it true that any continuous map $\mathbb{S}^n\to \mathbb{S}^m$ is not surjective if $n
Aspirin
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