Topology — Open Sets, Continuity & Key Concepts

Topology studies spaces through open sets and continuity instead of exact distance. The central definition you compute with is the continuity criterion: a function $f:X \to Y$ between topological spaces is continuous when

The symbols: $X$ is the source space, $Y$ the target, $U$ any open set in $Y$, and $f^{-1}(U)$ its inverse image in $X$. The fastest way into topology is to choose which subsets are open, then read continuity straight off that choice — no distance formula required.

Why this rests on open sets

Before the continuity test means anything, you need open sets. A topology on a set $X$ is a collection $\mathcal{T}$ of subsets, called open sets, satisfying:

1. $\varnothing$ and $X$ are in $\mathcal{T}$.
2. Any union of sets in $\mathcal{T}$ is also in $\mathcal{T}$.
3. Any finite intersection of sets in $\mathcal{T}$ is also in $\mathcal{T}$.

The pair $(X,\mathcal{T})$ is a topological space. Open sets encode the idea that a point has room around it. In the usual topology on $\mathbb{R}$, an open interval like $(1,3)$ is open because every interior point has a smaller interval around it still inside $(1,3)$. That local wiggle room is the heart of the subject, and it is exactly why the continuity criterion is built from inverse images of open sets: continuity means local structure is preserved. Openness is relative — the same subset can be open in one topology and not in another.

These open sets generate the rest of the vocabulary:

• Neighborhood: any set containing an open set around a point — local closeness without distance.
• Closed set: a set whose complement is open. On the real line, closed sets often contain their boundary, but that is a consequence, not the definition.
• Homeomorphism: a bijection that is continuous and has a continuous inverse. Homeomorphic spaces are "the same shape" topologically, but only because continuity holds in both directions — a bijection alone is not enough.

Worked example: why $f(x)=x^2$ is continuous in the usual topology

Take $f:\mathbb{R} \to \mathbb{R}$,

with the usual topology on $\mathbb{R}$. Pick the open set $(1,4)$ in the codomain and compute its inverse image:

which simplifies to

That set is open in the usual topology because it is a union of open intervals. Notice the direction the test runs: continuity is about inverse images of open sets being open, not about $f$ sending open sets to open sets. On $\mathbb{R}$, this condition matches the usual limit-based definition.

Try the computation yourself

Use the same $f(x)=x^2$, but start from the open set $(0,1)$. Compute $f^{-1}((0,1))$ and check whether it is open in $\mathbb{R}$. For a next step, compare the topological definition of continuity here with the limit-based one from calculus.

Computation traps to avoid

• "Open" is absolute. It is not — openness is defined relative to the chosen topology.
• Image vs. inverse image. Continuity uses inverse images of open sets. Images of open sets need not be open without extra conditions on $f$.
• Topology as relabeled metric geometry. Many spaces come from metrics, but topology keeps only the structure needed for local behavior and continuity.
• Every bijection is an equivalence. A homeomorphism must be bijective, continuous, and have a continuous inverse.

Topology supplies the language behind continuity, compactness, connectedness, and convergence in analysis, and reaches into geometry, dynamical systems, differential equations, and shape-based data analysis. For a beginner the payoff is concrete: continuity can be defined without coordinates or a distance formula.