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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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Is every rational sequence topology homeomorphic?

In the rational sequence topology, rationals are discrete and irrationals have a local base defined by choosing a Euclidean-converging sequence of rationals and declaring any cofinite subset of this sequence along with the irrational to be open. Do…
Steven Clontz
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When $X$ is homeomorphic to $X \times \mathbb{N}$?

Under what conditions $X$ is homeomorphic to $X \times \mathbb{N}$? where $\mathbb{N}$ is the discrete space.
Lo52
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Is the set $U(n,\mathbb R)$ of all upper triangular $n\times n$ matrices over $\mathbb R$ a connected set in $M(n,\mathbb R)?$

Is the set $U(n,\mathbb R)$ of all upper triangular $n\times n$ matrices over $\mathbb R$ a connected set in $M(n,\mathbb R)$ (with its usual topology after identification with $R^{n^2})?$ I think the answer is yes since connectedness is a…
Sriti Mallick
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Defining Open Sets on a topological space

Let $S$ be a set with continuously many elements, and $\_<\_:\_$ be a three-placed relation such that for any $x,y,z$ in $S$, $x
Lory
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Is the set $\{(x,y)\in\mathbb R^2:x^{2/3}+y^{2/3}\le1\}$ connected?

Is the set $\{(x,y)\in\mathbb R^2:x^{2/3}+y^{2/3}\le1\}$ connected? Please help me. I'm clueless. Added: Is the set convex?
Sriti Mallick
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Why is the sequential closure not sequentially closed?

In my understanding, a sequentially closed subset $A\subset X$ of a topological space $X$ is one that contains every sequential limit point of itself, whereas the sequential closure of $A$ is defined as $[A]_{seq}=\{x\in X: \exists (x_n)\in…
Bananach
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For which topological spaces $X$ is $X^n$ homeomorphic to the space of unordered n tuples of points in $X$

Consider a topological space $X$. Consider the spaces $A_n=X^n$ and $B_n=X^n/q_n$, the space of where $q_n$ is the equivalence relation where two points in $X^n$ are equivalent if one can be constructed from another via a permutation of the…
Mathew
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How many holes does my sock that has two extra holes have in it?

I have a pair of Nike Elite socks that have a hole on the heel and another hole at the ball of my foot. Here is the question: Topologically how many holes are in my sock? My friend and I have been arguing about this all night.
Mathematical Dummy
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Find an example of a complete bounded metric space which is not compact.

I think the infinite dimensional sphere satisfies the following criteria. However, I was hoping that someone could come up with a more elementary example. Thanks. Find an example of a complete bounded metric space which is not compact.
dunkindonuts
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Is there a continuous surjective map from $\mathbb{R}$ onto the subspace $\mathbb{R^{\infty}}$

Is there a continuous surjective map from $\mathbb{R}$ onto the subspace $\mathbb{R^{\infty}}$ of $\mathbb{R}^\omega$ with product topology, where $\mathbb{R^{\infty}}$denotes the set of points in $\mathbb{R}^\omega$such that all but finitely many…
Alex
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Continuous bijection from open n- ball to closed n- ball

Does there exist a continuous bijection from open n ball to closed n-ball? One with a simple argument can show that no such function exists for n=1.But, what about n>1?
Alex
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Partial Order induced topology

I was wondering if there is a canonical topology induced by a partial order on a set and how that relates to the total ordering topology (if it can be extended to a total ordering). I thought maybe the basis would be defined as in total orderings,…
user68193
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Let $Y \subset X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X \setminus Y$, then $Y \cup A$ is connected.

I am trying to solve Let $Y \subset X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X \setminus Y$, then $Y \cup A$ is connected. My ATTEMPT: We will show that $Y \cup A$ is connected. Let it is not, then there is…
user886636
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If $f:X\to Y$ is a one-to-one continuous mapping, and $Y$ is a $T_{2}$-space, then does $X$ too have to be a $T_{2}$-space?

Motivation: For any two points $y_{1},y_{2}\in Y$, there are disjoint open sets containing $y_{1}$ and $y_{2}$ separately, as $Y$ is a $T_{2}$-space. Say $f(x_{1})=y_{1}$ and $f(x_{2})=y_{2}$. Then, taking the inverses of the disjoint open sets, we…
user67803
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Complement of a Topology

Let $(X, \tau)$ topology, I was wondering, if given $$\tau'=\{A^C \mid A \in \tau\}$$ did $\tau'$ also a topology on $X$? If so, why? Thank you.
Blackoffe
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